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Handbook of Mathematics

Index

A-basis, 316 Adjoint functor, 302 A-module, 316 Adjoint functors, 595 Ab-category, 585 Adjoint functors (adjunction), 595 Ab-enriched (symmetric) monoidal category, 586 Adjoint map, 243 Abel lemma, 121, 453 Adjoint of a linear map, 243 Abel theorem, 129 Adjoint operator, 243, 375 Abel-Poisson, 487 Adjoint pair (of functors), 311 Abelian category, 584, 591 Adjointness, 589 Abelian group, 42, 43, 82, 84, 88, 89, 103, 474, 475, 516, 517, 521, 584, 590, Adjunction, 595 599 Adjunction isomorphism, 596 Abelian integral, 437 Adjunction of indeterminates, 120 Abelian semigroup, 43, 516 Admissible open set, 306 Abelian variety, 513 Admissible parameter, 393 Abelianization, 315 Admissible parameterization, 393 Abraham-Shaw, 748 Affine algebraic set, 513 Absolute complement, 8, 12 Affine bijection, 169 Absolute consistency, 5 Affine combination, 278 Absolute extrema, 45 Affine connection, 417, 418 Absolute frequency, 620 Affine coordinates system, 190 Absolute geometry, 149, 151, 152 Affine endomorphism, 191 Absolute gometry Affine form, 191 models 1,2,3,4, 151 Affine geometry, 155 Absolute homology groups , 291 Affine group, 168 Absolute Hurewicz theorem, 315 Affine independent, 278 Absolute maximum, 670 Affine independent set, 278 Absolute valuation, 79 Affine isometry, 187 Absolute value, 145 anti-displacement, 187 in the field of rationals, 145 displacement, 187 non-Archimedean, 145 Affine map, 168, 169 Absolute value of a rational number, 62 A ffi ne m ap in R3, 186 Absolutely convergent series, 324 Affine morphism, 564 Absolutely flat ring, 526 Affine n-chain, 293 Absorbing, 33 A ffi ne n-space, 270 Absorbing element, 12, 33 Affine neighborhood, 569 Absorbing on the left, 33 Affine orthogonal symmetry, 168 Absorbing on the right, 33 Affine plane, 155 Absorption, 8, 9 Affine plane of incidence, 155 Absorption laws, 12 Affine scheme, 270, 513, 563, 566 Abstract algebra, 263, 509 Affine set, 277 Abstract algebraic variety, 513 Affine set (convex set), 277 Abstract category, 38, 598 Affine space, 189, 277, 531 Abstract Riemann surface, 432 affine subspace, 190 Abstract simplicial complex, 295, 534 canonical structure, 190 Abstract space, 370 vectorial structure, 190 Abstract variety, 513 Affine space (algebraic geometry), 531 Acceptance region, 617 Affine space (analytic geometry), 531 Accumulation point, 26, 195, 198 Action-angle variable, 695 Affine subspace, 190 Acwise connectedness, 546 Affine subspace intersection, 190 Acyclic carrier, 304 Affine symmetry, 168 Acyclic complex, 290 Affine transformation, 169 Acyclic cover, 296 Affine variety, 264, 513, 531 Acyclic cover of a simplicial complex, 296 Affine Y-scheme, 564, 570 Acyclic free chain complex, 303 Affines map, 190 Acyclic free chain complex theorem, 303 Affinity, 168, 169 Acyclic graph, 260 A ffi x, 69, 71 Acyclic model, 303 Afraimovich-Shilnikov theorem, 739 Acyclic models, 303 Aggregate demand function, 677 Acyclic object, 556, 557 Agument predicate, 3 Acyclic resolution, 559 AIC and SC criteria, 647 Acyclic sheaf, 559 AIC criteria, 647 Acyclic space, 287 AIC criterion, 632 Aczel’s constructive Zermelo-Fraenkel (CZF), 16 A IC (p), 647 Addition and multiplication in N, 52 Aitken estimator, 633 A ddition in N, 52 Akaike criterion (AIC), 640 A ddition in Q, 61 Akaike function, 632 A ddition in Z, 58 Akaike information criterion, 632 Additive abelian group, 82 Akaike-Schwarz, 629 Additive associativity, 43, 516 Akaike-Schwarz criteria, 647 Additive category, 585 Alembert theorem, 224 Additive commutativity, 43, 516 A leph, 33 Additive functor, 303, 597 Aleph-0, 33, 35, 37 Additive group, 103 Aleph-1, 33, 35, 48 Additive group of integers modulo n, 87 Alexander-Veblen corollary, 296 Additive identity, 42, 43, 516 Alexander-Whitney theorem, 319 Additive inverse, 42, 43, 516 Alexandroff compactification, 210 Additive notation, 32, 39 Alexandroff extension, 210 additively and multiplicatively stable, 40 Alexandroff theorem, 225 Additively stable, 40 Algebra, xvii, 81, 133, 264 Additivity axiom, 553 associative, 133 Adequate, 297 commutative, 133 Adequate subcomplex, 297 of Banach, 442 Adherent point, 195, 198, 219, 674 σ-algebra (Σ-algebra, Ω-algebra), 511 Adherent point of a filter, 223 Algebra of polynomials, 107 Adic topology, 533, 569 Algebra of sets, 8 A djective Algebra of the disk, 460 autological, 13 Algebraic basis, 244 heterological, 13 Algebraic category, 593 Adjoint, 607 Algebraic closure, 69, 77, 124, 531 Adjoint endomorphism, 137 Algebraic closure of a field, 69, 78

7 7 9 Index

Algebraic closure of C, 77 Antidual, 405 Algebraic complex numbers over Q, 117 Antidual basis, 405 Algebraic curve, 150, 336 Antihomomorphism, 562 Algebraic dimension, 212 Antilinear map, 241 Algebraic dual, 238 Antinomy, 13 Algebraic element, 530 semantic or syntactic, 13 Algebraic elements of a field, 122 Antipodal map, 294 Algebraic extension, 123 Antipode, 271, 294 Algebraic extension of a field, 69, 531 Antisymmetric function, 138 Algebraic function, 335, 436 Antisymmetric map, 138 Algebraic fundamental group, 568 Antisymmetric multilinear form, 138 Algebraic geometry, 509 Antisymmetrization, 138 Algebraic K-theory, 269, 562 Antisymmetrization operator, 138 Algebraic number, 78, 117 Antisymmetry, 7 Algebraic number field, 78 Aperiodic solution, 722 Algebraic operation, 8, 510 Apollonius circle, 167 Algebraic operations in sets, 8 Apollonius definition of a circle, 168 Algebraic point, 437 Applied mathematics, xviii Algebraic problem, 129 Approximation, 331 Algebraic ramification, 433 Approximation by convolution, 491 Algebraic relation, 335, 336 Approximation by interpolation polynomial, 340 Algebraic representation of a complex number, 70 Approximation by polynomials Algebraic set, 513 on a compact of Cn, 236 Algebraic signature, 511 on a compact of Rn, 236 Algebraic space, 513 Approximation by trigonometric polynomials, 236 Algebraic structure, 38, 81, 512, 514, 582 Approximation methods, 354 of integers, 64 Approximation of vector functions, 236 of rationals, 65 Approximation theory, 338 of real numbers, 66 AR (autoregressive), 644 Algebraic structure of R, 321 AR model, 644 Algebraic surface, 150 AR(p), 634, 644 Algebraic system, 265, 280, 510 Arc, 196, 204 Algebraic topology, xvii, 247, 270, 509 Arc length, 358 Algebraic variety, 264, 513 Arc length of curve, 358 Algebraically closed, 77 Arc length parameterization, 499 Algebraically closed field, 78, 113, 531, 569 Arc of a curve, 392, 393 Algebras for a monad, 593 ARCH (AR conditional heteroscedasticity), 637 Algebrization, 129 ARCH model, 637 Almost everywhere convergence, 488 ARCH test, 637 Alternating algebra, 137 ARCH(q), 652 Alternating bilinear form, 449 Archimedean property, 52, 67 Alternating form, 138 Archimedean totally ordered commutative field, 66 Alternating group, 87, 138 Archimedean totally well ordered field, 78 Alternating map, 138 Archimedean valued field, 146 Alternating series, 324 Arcwise-connected, 545, 546 Alternative hypothesis H 1, 617 Arcwise-connected component, 204 Altitude, 166 Arcwise-connected set, 196, 204 Ambient manifold, 544 Arcwise-connected space, 204, 226, 545 Amenable group, 18 Area function, 168 Amnestic functor, 595 Area measure on a regular patch, 400 Ampère law, 364 Area of elliptic triangle, 179 Amplitude of cycle, 734 Area problem, 328 Amplitude spectrum, 662 Areolar function, 365 Analysis, xviii, 321 Areolar measure, 365 Analysis of autocorrelation function, 643 ARFIMA (AR fractionally integrated MA), 651 Analysis of residuals, 646 ARFIMA process, 651, 652 Analysis of shocks, 647 ARFIMA(p,d,q), 651, 652 Analysis of variance, 617 Argand-Cauchy plane, 71 Analysis of variance (ANOVA), 627 Argand-Gauss plane, 71 Analysis situs, xvii Argument, 2, 74, 455, 499 Analytic, 548 Argument of a complex number, 74 Analytic at, 548 Argument principle, 476 Analytic continuation, 421, 427, 441, 457 ariety (universal algebra), 513 Analytic continuations along different paths, 427 ARIMA (AR integrated MA), 644 Analytic function, 333, 415, 426, 433, 548 ARIMA model, 644 Analytic function at a point, 548 ARIMA process, 645 Analytic function on interior of domain, 548 Arithmetic, 51 Analytic functions, 421 Arithmetic function, 141 Analytic geometry, xvii, 130, 149, 181 Euclidean, 142 real or complex, 150 Arity, 510, 511 Analytic manifold, 540 ARMA (autoregressive moving average), 645 Analytic representation, 172 ARMA model, 645 Analytic set, 560 ARMA(p,q), 645 Analytic space, 560 A R M A X , 646 Analytic subspace, 560 Arnold, 687, 695 Analytic variety, 513 Arnold tongue, 742 Andronov-Hopf bifurcation, 735, 748 Arnold tongues, 740 Andronov-Hopf theorem, 735 Arrangement, 613 Andronov-Pontryagin theorem, 724 Arrangements with repetitions, 614 Andronov-Witt theorem, 719 Arrangements without repetition, 614 Angle, 160, 163 A rrow , 576 acute, 160 Arrowsmith-Place, 708, 712 obtuse, 160 Articulation vertex, 262 of opening, 184 Artin braid group, 539 refl ex, 160 Artinian ring, 525 right, 160 Artinian set, 35 straight, 160 Ascending chain condition (ACC), 523 Angle measure, 191 Ascoli theorem, 236, 442 Angle measure on a regular patch, 400 Associate, 309 Angle measurement, 161 Associative law, 41 Angle trisection, 75, 130 Associative magma, 41, 515 Angle-preserving transformation, 438 Associativity, 8, 9 Annihilating element, 12 Associativity isomorphism, 587 Annulus, 195, 428, 447, 450, 464 Associator, 586 A nosov, 705 Astroid, 336 Anosov automorphism, 703 Asymptote, 334 Anosov diffeomorphism, 705, 732 Asymptotic behavior of Fourier coefficients, 655 ANOVA equation, 627 Asymptotic line, 403 ANOVA table, 617 Asymptotic point, 381 for simple regression model, 627 Asymptotic right triangle, 178 Antecedent, 30, 47, 344, 620 Asymptotical stability, 719 Anti-adjoint, 243 Asymptotically stable, 687 Anti-matched lines, 153 Atlas, 413—415, 542, 687 Anti-matching, 152 Atlas of class Cr (or Cr -atlas), 414 Anti-matching theorem, 153 Atom , 2, 10 Anti-paired lines, 153 Atom of a lattice, 10 Anti-self-adjoint operator, 243 Atomic formula, 16 Antiautomorphism, 562 Attaching cell, 299 Anticonformal mapping, 438 Attaching map, 298, 299, 535 Antiderivative, 349, 352, 353 Attracting or repelling basin, 747

780 Index

Attractor, 715, 724 of number system, 78 Attractor in sense of Ruelle (1981), 733 Axiomatic method, 5, 149, 150 Attractor in the sense of Milnor, 715 Axiomatic set theory, 12—15, 577 augmentation, 303 Axiomatic system, 3 Augmentation homomorphism, 553 A x iom s, 1 Augmentation preserving, 303 Euclid, 149 Augmented complex, 296 of Zermelo-Fraenkel set theory, 16 Augmented Dickey-Fuller test (ADF), 644 of ZFC , 16 Augmented singular complex, 292 separation, 205, 206 Aureum theorem, 144 T0,T1,T2,T3,T4, 206 Autocorrelation function, 643, 725 Axioms of categories, 590 Autocorrelation function of sampling, 643 Axioms of closed sets, 218 Autocorrelation of errors, 633 Axioms of Zermelo-Fraenkel set theory, 16 Autocorrelation test of errors, 639 Axis of curvature, 396 Autofunctor, 595 Axis reflection, 159 Autological adjective, 13 Axonometric projection, 174 Automorphism, 33, 38, 99, 483, 520, 598 Automorphism group, 105 b-adic number, 68 Automorphism set, 105 Bézout coefficients, 60, 111 Bézout identity, 60 Automorphisms of C, S2,D, 483 Autonomous differential equation, 716 Bézout theorem, 60 Autonomous linear differential equation, 716 Bachmann, 151, 157 Bachmann metric plane, 151 Autonomous system, 693 Autoregressive linear model, 639 Back face, 318 Backward elimination, 632 Axial vector, 184 Baire category theorem, 232, 724 A xioM Baire function, 368, 369 of countable choice, 19 Baire property, 232 Axiom, 2, 5, 9 Baire space, 232 of extensionality, 17 Baker map, 732 A,A* (bisector), 159 B all, 50 A1,A2,A3 (affine incidence plane), 155 Banach algebra, 24, 442, 494 C (connection), 154 Banach fixed point, 372 D (completion) Dedekin, 160 Banach fixed-point theorem, 372 functional , 14 Banach functional, 20 H (hyperbolic), 154 Banach space, 20, 21, 237, 241, 371, 489 Hausdorff, 219 Banach theorem, 241, 391 M1,M2,M3 (metric plane) Absolute geo., 151 Banach-Alaoglu theorem, 442 of Cartesian product, 12 Banach-Steinhaus theorem, 241, 371 of choice, 14, 16, 17 Banach-Tarski paradox, 8, 17 of comprehension, 14, 16 Barratt-Whitehead lemma, 292 of denumerable choice, 19 Barrow rule, 499 of dependent choice, 19 Barycenter, 190, 252, 278, 675 of existence, 14 Barycentric coordinates, 190, 252, 278 of extensionality, 10, 11, 14, 16 Barycentric subdivision, 293 of foundation, 16, 17 Barycentric subdivision of an affine n-simplex, 293 of induction, 50 B ase, 560 of infinity, 14, 16, 17 Base for a topological space, 537 of pairing, 11, 16, 17 Base line, 157 of regularity, 16, 17 Base of fiber space, 562 of replacement, 16, 17 Base scheme, 563 of separation, 11, 16 Base space, 306, 560, 574 of singletons, 10, 14 Base ten numeral system, 56 of subsets, 16 Base-10 numeral system, 56 of Tarski, 16 Basepoint, 214 of the empty set, 14 Basepoint preserving map, 272 of the pair, 11 Basic dynamics of diffeomorphisms, 687 of the power set, 16, 17 Basic variable, 670 of the set of parts of a set, 11 Basin of attraction, 715, 724, 733 of the set of the parts of a set, 14 Basis, 102, 199, 405, 519 of the sum of set, 17 Basis for a topology, 308 of the sum set, 16 Basis of a K-vector subspace, 108 of the unordered pair, 16, 17 Basis of a topology, 199 of union, 11, 14, 16, 17 Basis of neighborhoods, 200, 218, 222 P (polar trilateral), 154 Basis of open sets, 218 P1,P2,P3 (projective incidence plane), 155 B auer, 370 P5,P5’ (Fano), 158 Bayes theorem, 619 R (rectangle), 153 Behaviors of second-order linear systems, 716 schema of collection, 17 diagram of behaviors, 717 schema of replacement, 17 interpretation of eigenvalues , 717 schema of restricted comprehension, 17 B elote, 615 schema of specification, 17 Bernoulli inequality, 323 T0,T1,T2,T3,T4, 206 Bernoulli number, 337 T 2, 199 Bernoulli shift map, 725 Axiom (postulate), 5 Bernstein polynomial, 338 Axiom of bisector (A),(A*), 159 Bessel inequality, 244, 653 Axiom of Cartesian product, 12 Best approximation, 338, 394 Axiom of choice, 14, 16, 22, 25, 32, 579 Best contact, 394 Axiom of comprehension, 14, 16 Beth’s definability theory, 5 Axiom of connection (C), 154 Betti group, 257, 550 Axiom of countable choice, 19 Betti number, 257, 412 Axiom of Dedekin (D), 160 Bezout identity, 143 Axiom of denumerable choice, 19 Bezout relation, 120 Axiom of dependent choice, 19 Bezout theorem, 60 Axiom of existence, 14 Bezout theorem (for polynomials), 111 Axiom of extensionality, 10, 11, 14, 16, 17 Bi-ratio, 171 Axiom of extensionality (for maps), 30 Bianchi identity, 419 Axiom of foundation, 16, 17, 578 Bicategory, 585, 593 Axiom of infinity, 14, 16, 17, 37 bicategory, 264 Axiom of pairing, 11, 16, 17 Biclosed monoidal category, 586 Axiom of polar trilateral (P), 154 Bicomplete category, 585 Axiom of regularity, 16, 17 Biconnected graph, 262 Axiom of replacement, 16, 17 Bidual, 239, 405 Axiom of revealed preferences, 760 Bidual of a normed vector space, 239 Axiom of separation, 11, 16, 31 Bienaymé-Chebychev inequality, 620 Axiom of singletons, 10, 14 Bienaymé-Tchebychev inequality, 620 Axiom of subsets, 16 Bifunctor, 264, 585, 586, 595 Axiom of the empty set, 14 Bifunctor (binary functor, i.e. 2-ary), 595 Axiom of the pair, 11 Bifurcation, 733, 734, 748 Axiom of the power set, 16, 17 Andronov-Hopf, 735, 748 Axiom of the set of parts of a set, 11 Andronov-Hopf supercritical, 749 Axiom of the set of the parts of a set, 14 cusp, 712 Axiom of the sum of set, 17 fl ip, 748 Axiom of the sum set, 16 fold, 748 Axiom of the unordered pair, 16, 17 H opf, 733 Axiom of union, 11, 14, 16, 17 Hopf subcritical, 712 Axiom schema of collection, 17 Hopf supercritical, 712 Axiom schema of replacement, 17 node-saddle, 733 Axiom schema of restricted comprehension, 17 of fixed points, 733 Axiom schema of specification, 17 of periodic orbits, 734 Axiomatic construction period-doubling, 748

781 Index

pitchfork, 733 Bogdanov-Takens bifurcation, 736 saddle node, 718, 748 B olyai, 169 subcritical, 734 Bolzano-Weierstrass property, 210 subharmonic, 734 Bolzano-Weierstrass theorem, 322, 323 supercritical, 733 Boolean algebra, 19 transcritical, 733 Boolean algebras, 9 Bifurcation diagram, 713, 735, 736, 749, 751 Boolean lattice, 9, 10, 618 Bifurcation diagram of Lorenz model, 751 Boolean prime ideal theorem (BPI), 19, 23, 24 Bifurcation of double periodic orbits, 736 Boolean ring, 10, 93 Bifurcation value, 734 Bordant, 373, 547 Bifurcations in Morse Smale systems Bordant manifolds, 547 global bifurcation, 737 Bordism, 263, 373, 549 saddle-node separatrix disappearance, 738 Bordism group, 547 saddle-node to periodic orbit, 738 Borel measure, 725 local bifurcation nearby periodic orbits B orel σ-algebra, 618 bifurcation of double periodic orbits, 736 Borel set, 618, 725 centre manifold theorem for maps, 736 Borel sigma-algebra, 618, 725 creation of a torus, 737 Borel-Carathéodory inequality, 467 period-doubling (flip bifurcation), 737 Borel-Lebesgue, 223 local bifurcations nearby steady states, 734 Borel-Lebesgue property, 223 Bogdanov-Takens bifurcation, 736 Borsuk theorem, 473 breaking symetry, 736 Borsuk-Ulam, 294 centre manifold theorem, 734 Borsuk-Ulam theorem, 320 cusp bifurcation, 735 Boundary, 286, 294, 547 generalized Hopf bifurcation, 736 Boundary conditions, 391 Hopf bifurcation, 735 Dirichlet, 391 saddle-node, transcritical bifurcations, 734 Neumann, 391 Bifurcations of fixed points, (Cod-1), 733 R obin, 391 subcritical bifurcations, 734 Boundary map, 553, 558 supercritical bifurcations (normal form) Boundary morphisms, 597 H opf, 733 Boundary of a singular n-simple, 286 pitchfork, 733 Boundary operator, 286, 534, 550 saddle-node, 733 Boundary point, 195, 294 transcritical (stability exchange), 733 Boundary value problem, 378 Bifurcations theory in Morse Smale systems, 734 B ounded, 21 Big étale site, 568, 569 Bounded distance, 217 Biggest element, 9 Bounded from above, 46 Biharmonic oscillator, 695 Bounded function, 21 Biholomorphic map, 438 Bounded lattice, 23 Biholomorphism, 482, 483 Bounded linear functional, 19, 21 Biholomorphism,(conformal map), 482 Bounded linear operator, 372 Bijection, 33, 37, 520 Bounded part, 209, 216 inverse, 36 Bounded tile, 209 Bijective homomorphism, 520 Bounded variation, 653 Bijective homomorphism (bimorphism), 520 Bouquet of circles, 298 Bijective map, 29, 31, 519 Bouquet of circles (rose), 298 Bijective map defined via compositions, 30 Bourbaki, 205 Bijective morphism, 33 Bourbakist conception of mathematics, 37 Bijective morphism (isomorphism), 33 Boussinesq, 749 Bijective sets, 29 Box-counting dimension, 730 Bijectivity of Galois map, 127 Box-counting dimension properties, 730 Bilinear extension, 103 Box-Jenkins method, 645 Bilinear form, 406 Box-Pierce statistic, 643 Bilinear function, 304 B ra, 409 Bimorphism, 520, 598 Brachistochrone, 372, 373 Binary functor, 264 Bracket of two vector fields, 416 Binary numeral system, 56 B raid, 539 Binary operation, 265, 510 Braid group, 539 Binary relation, 7, 13, 27, 28 Braid index, 539 antisymmetric, 28 Braided category, 587 asymmetric, 28 Braided monoidal category, 587 biective (or biunivocal), 28 Braiding, 587 doubly surjective, 28 Branch, 423, 431 injective, 28 Branch cut, 423, 431 left surjective, 28 Branch point, 422, 431, 433 properties, 28 Breaking symetry, 736 reflexive, 28 Breit-Wigner curve, 663 right surjective, 28 Breusch-Godfrey test, 634 symmetric, 28 Broomhead-King, 416 total, 28 Brouwer, 4, 212, 215, 472, 674, 677 transitive, 28 Brouwer degree, 413 univocal, 28 Brouwer fixed point theorem, 215, 677 Binomial coefficient, 90, 613 Brouwer theorem, 214, 472 Binomial distribution, 617, 621 Brownian motion, 651 Binomial equation, 129 Bubble sort, 87 Binomial expansion, 90 Bundle, 153, 263, 268, 561 Binomial formula, 92 Bundle (category), 561 Binomial series, 90, 333 Bundle (topology), 561 Binomial theorem, 90, 92 Bundle rank, 269, 562 Binormal vector, 394 Buniakowski inequality, 191 Bipartite Graph (or bigraph), 259 Burali-Forti, 13 Biperiodic dynamics, 722 Burali-Forti paradox, 49 Bipoint, 161 By recurrence, 5 Bipolar rectangular impulse, 663 C Biquadratic reciprocity theorem, 145 is algebraically closed, 115 Birational transformation, 150 ∞ partition of unity subordinate to covering, 499 C Birkhoff ergodic theorem, 725 -analytic function, 453, 457 C-C function, 423 Birkhoff HSP theorem, 514 É Birkhoff prime ideal theorem, 23 C-C function, 423 É Birkhoff theorem, 514, 718 C-C function, 423 É É Birth of an invariant torus, 748 C-C-function, 423, 425 Bisector, 152 C-differentiable function, 424, 452, 548, 549 Bisector axiom (A),(A*), 159 C∞-manifold, 541 Bite vector, 32 C∞-structure, 541 Bivalence principle, 1 partition of unity, 495, 500 C∞ Bivalent logic, xvii C∗-algebra, 562 Blakers-Massey theorem, 315 Cr-manifold, 687 Blaschke condition, 464 Caeteris paribus, 647 Blaschke products, 468 C ah, 175 B lo ck, 262 Calculation Block-Guckenheimer-Misiuriewicz theorem, 733 of geodesic curvature, 401 Blowing-up, 701 of normal curvature, 401 directional, 702 Calculus and analysis, xviii p olar, 701 Calculus of variations, 372 BLUE, 634, 635 Cancellable, 40 BLUE (Best Linear Unbiased Estimator), 629 Cancellable element, 33, 40 Blue sky catastrophe, 746, 748 Cancellation property, 40 Bodganov, 712 Canonical basis, 191, 443 Bogdanov, 712 Canonical basis of the K-vector subspace, 108 Bogdanov, Takens, 712 Canonical euclidean norm, 191

782 Index

Canonical flabby resolution, 559 Cauchy formula (Homotopy form), 471 Canonical flasque resolution, 559 Cauchy inequalities, 457, 486 Canonical form of a circle map, 740 Cauchy integral Canonical injection, 30, 520 formula, 425 Canonical map, 30, 201, 563 property, 425 Canonical morphism, 87 Cauchy integral formula, 425, 429 Canonical projection, 223, 599, 603 Cauchy integral property, 425 Canonical representation of simple arc, 395 Cauchy integral theorem, 425, 429, 660 Canonical resolution, 559 Cauchy integration criterion, 355 Canonical scalar product in vector space Rn, 191 Cauchy kernel, 460 Canonical structure, 190 Cauchy kernel of unit disk, 460 Canonical surjection, 29, 34 Cauchy mean-value theorem, 331 Canonical topology, 609 Cauchy principal value, 661 Cantor, 13, 37, 47, 66 Cauchy rational sequence, 69 Cantor comb, 730 Cauchy real sequence, 69 Cantor diagonal method, 32, 34 Cauchy sequence, 21, 66, 67, 146, 372 Cantor diagonalization, 33 rational, 69 Cantor function, 740 real, 69 Cantor native language, 35 Cauchy theorem, 456, 474 Cantor sequence, 49 Cauchy theorem (Homology version), 474, 475 Cantor set, 706, 709, 730, 731 Cauchy theorem (Homotopy version), 470, 474 Cantor theorem, 31, 37, 228 Cauchy tranform, 459 Cantor’s transfinite numbers, 48 Cauchy transform, 453 Capacity dimension, 730 Cauchy transform of a Radon measure, 453 Capital per unit of labor, 760 Cauchy transform of Borel measure, 459 Carathéodory theorem, 485 Cauchy-Goursat theorem, 453, 461 Cardan formula, 74 Cauchy-Kowalewska theorem, 389 Cardan method, 128 Cauchy-Pompeiu formula, 456, 489 Cardinal, 10, 33, 34, 48 Cauchy-Riemann conditions, 421, 424 addition, 34 Cauchy-Riemann equation, 457 arithmetic, 34 Cauchy-Schwarz inequality, 191 comparison, 34 Causality in the sense of Granger, 647 exponentiation, 34 Causality in the sense of Sims, 647 multiplication, 34 Cavalieri, 357 of a set, 36 Cavalieri principle, 357 of finite or infinite set, 34 Cavalieri theorem, 357 of reals, 37 Cayley-Menger determinant, 251, 255 of the set of reals, 35 Cconnectedness properties of topological spaces, 538 operations, 34 Cech cohomology, 550, 558, 559 Cardinal number, 10, 33 Cech cohomology group, 558 Cardinal of a group, 82 Cech complex, 550 Cardinality, 33 C eiling, 61 Cardinality 1, 33 Ceiling function, 61 Cardinality 2, 33 Cell, 534, 585 C arnap, 1 Cell (algebraic topology), 534 Carrier (universe), 510 Cell (geometry), 534 Carrier function, 303 Cellular, 301, 535 Carrier set, 510 Cellular approximation, 302 Carrier set (underlying set), 510 Cellular approximation theorem, 302 Cartan, 138, 205 Cellular chain complex, 301 Cartesian closed category, 588 Cellular chain complex lemma, 301 Cartesian geometry, 130 Cellular chain complex of a filtration, 301 Cartesian product, 13, 27 Cellular filtration, 301 Cartesian representation of a complex number, 70 Cellular homology, 301 Cascade of period-doublings, 738 Cellular map, 301 Casorati-Weierstrass theorem, 464 Cellular space, 301 C at, 584 Cellular theory, 287 Catalan numbers, 32 Center, 83, 717 Catastrophic instability, 749 Center (centre), 83 Categorical direct sum, 603 Center manifold, 700, 748 Categorical product, 603 Center manifold theorem, 700 Categorical sum, 603 Center of the sheaf, 157 Categories of rational functions, 335 Centering parameter, 666 Category, 38, 256, 265, 270, 272, 576, 577, 581, 582, 598 Central angle, 163 by Eilenberg-Mac Lane, 576 Central limit theorem, 617 by Mac Lane, Buchsbaum, Grothendieck, Heller, 577 Central projection, 170 Category Ab (AbGrp), 583 Central projections in P 3 (R), 170 Category AbGrp (or Ab), 583 Central projections in R3, 170 Category associated with ordered set, 589 Central sheaf, 157 Category Cat, 583 Centre, 83, 381 Category Comp, 289, 290 Centre manifold, 748 Category CPO, 583 Centric, 156 Category Diff, 583 Centric perspective, 156 Category Diff b, 583 Ceteris paribus, 647 Category Fields, 583 C evian, 166 Category Grp (Groups, Group), 583 C H , 15 Category Heyt, 583 Chain, 21, 36, 44, 45, 260, 285 Category hoTop, 583 Chain complex, 289, 533, 550, 589 Category isomorphism, 594 Chain complex of finite type, 318 Category Man, 583 Chain equivalence, 290, 303 Category Man b, 583 Chain equivalent, 290 Category Mon, 583 Chain group, 301 Category of (Eilenberg-Moore) algebras, 594 Chain homotopic, 290 Category of all étale morphisms, 568 Chain homotopic maps, 290 Category of all abstract simplicial complexes and simplicial m aps, 295 Chain homotopy, 290 Category of chain complexes, 589, 592 Chain map, 290 Category of functors, 586 Chain of an ordered set, 36 Category of schemes, 270, 563 Chain of fields, 129, 130 Category of sheaves of sets, 551, 569 C haitin, 3 Category of small categories, 584 Change of admissible parameter, 393 Category PoSet, 583 Change of basins, 747 Category PreOrd, 583 Change of basis, 405 Category Rings, 583 Change of coordinate system, 187 Category Sets, 583 Change of coordinates, 414, 542 Category theory, 265, 575 Change of local coordinates, 541 Category Top, 583 Change of parameterization, 499 Category Vec, 583 Change of variable, 353, 380 Category Vec C, 583 Change of variables, 357 Category Vec R, 583 Chaos, 714, 732 Category with a generator, 591 (DO) density of orbits, 732 Catenary ring, 525 (SD IC ), 732 Catenary scheme, 564 (TT) topological transitivity, 732 Cathetus, 178 Chaos in the sense of Devaney, 732 Cauchy convergence criterion, 323 Chaos in the sense of Li-Yorke, 732 Cauchy criterion, 228, 244 Chaos theory, xviii Cauchy filter, 231 Chaotic attractor, 732 Cauchy form of Taylor remainder, 332 Chaotic dynamical system, 732 Cauchy formula, 425, 456 Chaotic systems in the sense of Devaney, 732 for a disk (direct demonstration), 459 Character of a point, 538 Cauchy formula (Homology form), 475 Character of a topological space, 538

783 Index

Characteristic, 94, 530 Closure , 674 Characteristic class of a fibration, 320 Closure (abstract algebra), 520, 521 Characteristic equation, 385 Closure of valued fields, 146 Characteristic function, 32 Closure operator, 102 Characteristic map, 300, 535 Closure theorems, 155 Characteristic of a ring, 94 Cluster point, 26, 195, 220 Characteristic polynomial, 135 of a sequence, 322, 323, 341, 423 Characteristic root, 133 Cluster point of a map, 221 Characteristic subspace, 135 Cluster point of a sequence, 221 Characteristic value, 133 Cn[a, b] space, 371 Characterization Co-cone, 601 of simply connected open sets, 485 Co-unit, 320 Characterization of an attractor, 730 Co-unit of a graded co-ring, 320 fixed point: Lyapunov exp. sign (-,-,-) , 730 Cobordant, 547 limit cycle: Lyapunov exp. sign (0,-,-) , 730 Cobordant manifold, 547 strange attractor: Lyap. sign (+,0,-) , 730 Cobordism, 373, 547 Torus: Lyapunov exp. sign (0,0,-) , 730 Cobordism class, 547 Characterization of fields by ideals, 101 Cobordism group, 547 Chart, 413, 541, 542, 687 Coboundary, 317, 534 Chasles relations, 75 Coboundary map, 555 Chebychev inequality, 620 Coboundary map (differential), 555, 556 Chebyshev (Tchebychev, Tchebicheff, Ceby˘sev,˘ Tschebyscheff, Chebishev), Coboundary operator, 534, 550 75, 339 Coboundary operator (codifferential), 558 Chebyshev approximation, 669 Cobounding, 373 Chebyshev distance, 220 cochain, 317 Chebyshev norm, 338, 340 Cochain complex, 534, 555, 556 Chebyshev norm (uniform norm), 338 Cochrane-Orcutt method, 635 Chebyshev polynomials, 75, 116, 338 Cocomplete category, 305, 585 Chebyshev polynomials of the first kind, 116 C o cone, 601 Chebyshev polynomials of the second kind, 116 Coconstant morphism, 599 Chi-squared distribution, 617 Cocycle, 317, 534 Chi-squared test, 617, 622 Codifferential, 558, 559 Chillingworth, 687 Codimension, 699, 712, 734 Chinese congruence, 97 Codimension of a bifurcation, 736 Chinese hypothesis, 97 Codimension-1, 733 Chinese remainder theorem, 97 Codomain, 14, 29, 30, 38, 519 Chinese theorem, 97 Coefficient group, 302 Choice function, 16, 17, 19 Coefficient of degree k, 108 Choice of optimal model, 631, 632 Coefficient of determination, 627, 629 Chordal distance, 423 Coefficients, 304 Chordal theorem, 167 Coequalizer, 305, 588, 604 Chow test, 635 Coexact sequence, 313 Christoffel symbol of the first kind, 418 Cofactor, 106 Christoffel symbols, 417 Cofibration, 314 Christoffel symbols of the second kind, 418 Cofinal, 15, 23, 578 C hurch, 3 Cofinality, 23 Circle, 163, 184, 185 Cogroup object, 310 Circle diffeomorphism, 692 Cohen-Macaulay ring, 525 Circle diffeomorphisms with rational number, 692 Cohen-Macaulay scheme, 564 Circle map, 742 Coherence conditions, 593 Circle of curvature, 177, 396, 398 Coherence property, 587 Circle of curvature for an ellipse, 397 Coherent sheaf, 553 Circle quadrature, 75, 131 Cohomology, 263, 316, 476, 549, 556 Circle squaring, 75, 131 Cohomology (of topological space), 550 Circuit, 260, 297 Cohomology (with values in sheaf), 550 Circular average, 491 Cohomology class, 317 Circular function, 174, 336 Cohomology functor, 597 Circular permutation, 91 Cohomology group, 263, 316, 317, 476, 550 Circumcenter, 167 Cohomology groups, 568 Circumcircle, 166 Cohomology ring, 318, 319, 550 Circumradius, 167 Cohomology theory, 263, 476, 509 Ck-atlas, 541 Cohomology with compact support, 571 Class of algebras, 594 Cohort data, 625 Class of generators for a category, 581 Coincident transformation, 161 Class of groups, 317 Cointegrated series, 648 Class V, 15, 578 Cointegration, 626, 648 Classes of quadrics, 188 Cointegration analysis, 648 Classical harmonic analysis, 487 Cointegration rank, 649 Classification Cointegration relation, 650 of quadrilaterals and triangles, 170 Cointegration relation test, 650 Classification of closed surfaces, 255 Cointegration test, 648 Classification of fields, 125 Cointegration theory, 626 Classification of metric planes, 155 Cointegration vector, 648 Classification of patch points, 402 Cokernel, 599, 605 Classification of periodic orbits, 719 Colimit, 305, 602 Classification of real functions, 328 Collection of morphisms, 38, 598 Classification of valued fields, 145 Collectivizing property, 11 Classifying morphism, 581 Collectivizing relation, 8 C lique, 262 Collinear, 152 Clopen set, 537 Collinear vectors, 181 Closed 1-form, 316 Collinearity, 181 Closed arc, 196 Collineation, 151, 156, 171 Closed ball, 216 orthogonal, 151 Closed curve, 214, 545 projective, 149 Closed curve contractility, 546 Collineation of a projective plane, 156 Closed disk, 71, 196 Collision, 748 Closed Eulerian chain, 260 Coloring, 214, 261 Closed form, 448 Comatrix, 106 Closed graph theorem, 241, 372 Comb space, 203 Closed immersion, 564, 571 Combination (or selection), 613 closed immersion, 570 Combination with repetitions, 615 Closed Jordan curve, 197 Combinations without repetition, 615 Closed k-cell, 294 Combinatorial analysis, 54, 613 Closed manifold, 542 Combinatorial homotopy, 254 closed manifold, 547 Combinatorial theory, 613 Closed manifold (compact boundaryless manifold), 547 Combinatorial topology, 250, 254 Closed monoidal category, 586 Combinatorics, 254, 613 Closed orbit, 722 Comeagre set, 232 Closed part, 216 Comma category, 580, 584 Closed path, 279 Comma derivative, 417 Closed set, 26, 195, 198 Common divisor , 57 Closed subscheme, 564 Common multiple, 58 Closed surface, 255 Commutative Abelian groups, 89 classification, 255 Commutative and unital ring, 522 Closed surfaces, 255 Commutative and unitary ring, 522 Closed under, 520, 521 Commutative Banach algebra, 24 Closed-bounded, 197 Commutative diagram, 30, 273, 600 Closed-bounded property, 197 Commutative field structure, 64 Closure, 24, 27, 42, 195, 216, 520, 521 Commutative group, 82, 89, 475 of a connected part, 226 Commutative group (Abelian group), 82

784 Index

Commutative group for the composition law, 156 Complex representation of Fourier sum, 656 Commutative ring, 522 Complex sequence, 423 Commutative ring with identity, 522 Complex sphere, 78 Commutative ring with unity, 522 Complex structure, 433 Commutative semigroups, 92 Complex vector bundle, 269, 561 Commutative unit ring, 10, 64 Complex vector space, 20 with no divisor of zero, 43, 516 Complex-differentiable, 549 Commutativity, 8, 9 Complex-differentiable (holomorphic, analytic), 549 Commutator, 315, 502 Complex-valued function, 421 Commutator subgroup, 315 Composition Comonad, 593 of reflections, 153 Comonad (cotriple), 593 Composition law, 29, 30 Compact, 197 associative, 30 Compact boundaryless manifold, 547 external, 38, 43, 516 Compact boundaryless manifold (closed manifold), 547 internal, 38, 39, 41, 42, 515 Compact invariant chaotic set, 732 Composition of differentiable functions, 330 Compact manifold, 541, 547 Composition of functions, 344 Compact operator, 375 Composition of homotheties, 165 Compact polyhedral pair, 302 Composition of maps, 30 Compact set, 26, 197, 208, 209, 715 Composition of reflections, 153 Compact space, 26, 78, 208, 209, 223, 538 Composition of relations, 29 by adjunction of a point (at infinity), 78 Composition of rotations, 41, 515 product, 224 Composition operators, 493 Compact subset, 197 Composition rule, 330 Compact support, 288, 668 Compositum, 162, 530 Compact support theorem, 288 Computation costs of Fourier coefficients, 656 Compact-open topology, 209, 309 Computation of complex Fourier coefficients, 656 Compact-open topology on XY , 309 Comultiplication, 312 Compactification, 78, 209, 210, 422, 571 Concave function, 677 of the plane, 78 Concentrated probability measure, 725 Compactification of C, 421, 422 Concept of convergence, 38 Compactness, 208 Concept of correlation, 625 Compactum, 210 Concrete category, 582, 584 Comparability, 21, 44, 579 Concrete Riemann surface, 432 Comparable distances, 217 Concurrent lines, 152 Comparison of Fourier and Laplace transforms, 663 Condensation point, 195 Comparison of topological spaces, 200 Conditional distribution, 618 Comparison of topologies, 199, 218 Conditional expectation, 618 Compass, 129 Conditional forecast, 630 Compatibility, 414 Conditional probability, 615, 618 of a map with topological structures, 414 Conditionally compact set, 375 Compatibility of charts, 414 Conditions of order (of identifiability), 642 Compatibility of charts of an atlas, 432 Conditions of rank, 642 Compatible chart, 541 Cone, 184, 276, 601 Compatible charts, 542 Cone construction, 288 Complement, 8, 12, 13 Cone of revolution, 172, 184, 185 Complement formula, 468, 477 Cone over X, 276 Complementary angles, 162 Confidence interval, 616, 629 Complementary set, 195 of the variance of error, 629 Complementation, 9 Confidence level, 616 Complemented lattice, 9, 23 Configuration, 156 Complete algebraic variety, 513 Conformal map, 438, 482 Complete axiomatic set theory, 13 Conformal mapping, 421, 438, 482 Complete category, 305, 585 Conformal transformation, 421, 438, 482 Complete field, 322 complex plane, Riemann sphere, 438 Complete graph, 259 from C to C, 439 Complete lattice, 128 from C to C, 439 É É Complete local ring, 569 from the interior unit disk to itself, 439 Complete metric space, 67, 215, 228, 230 of simply connected domain, 439, 440 complete normed linear space, 21 Conformally equivalent, 482 Complete normed space, 20 Congruence, 29, 143, 151, 161, 273 Complete normed vector space, 21, 237, 241, 371 Congruence modulo an ideal, 143 Complete pre-Hilbert space, 137, 371 Congruence theorems, 163 Complete quadrangle, 158 Congruence transformation, 149 Complete quadrilateral, 158 Congruent, 143 Complete space, 422 Congruent figures, 9 Complete uniform space, 231 Congruent sets, 160 Completed local ring, 569 Conic, 185, 186 Completely regular space, 207 Conic section, 156, 185 Completion, 78, 160, 215, 528 Conic section directrix, 185 of a metric space, 229 C onics, 184 Completion (abstract algebra), 528 Conjecture, 2, 5 Completion of a metric space, 229 Conjugacy, 690, 723 Completion of a ring, 528 Conjugacy class, 86 Completion of metric space, 215 Conjugacy of diffeomorphisms, 690 Completion of pre-Hilbert space, 242 Connected, 196, 538, 539 Completion of uniform space, 231 Connected acyclic graph, 260 Completion process, 78 Connected compact set, 214 Complex, 289 Connected component, 203, 226, 262, 545 Complex analysis, 421 Connected graph, 260 Complex analytic function, 548 Connected Hopf algebra, 320 Complex analytic function (holomorphic function), 548 Connected set, 195, 196 Complex analytic funtion, 548 Connected sinplicial complex, 297 Complex analytic manifold, 540 Connected space, 197, 202, 203, 225, 545 Complex conjugate, 71, 76 Connected subset, 203 Complex continuation, 421 Connectedness, 195 Complex curvilinear integral, 425 Connectedness (connectivity), 539 Complex derivability, 421 Connecting homomorphism, 291, 554, 560 Complex differentiability, 424, 548, 549 Connecting morphism, 597 Complex eigenvalue, 686 Connecting morphism (boundary morphism), 597 Complex entire series, 426 Connection, 417 Complex exponential, 76, 454 Connection coefficient, 417, 418 Complex exponential function, 72, 76 Connection curvature, 417 Complex Fourier coefficients, 656 Connection geodesic, 417 Complex function, 423 Connection on a differentiable manifold, 417 Complex Hahn-Banach theorem, 25 Connection torsion, 417 Complex infinity, 78 Connections, 1 Complex line bundle, 561 Connections of propositions, 1 Complex measure, 442 Connectives Complex number, 69 and, 1 addition, 69 equivalent, 1 calculation rules, 76 im plies, 1 multiplication, 69 nand, 1 normal representation, 76 non-equivalent, 1 Complex plane, 71, 216, 717 nor, 1 homeomorphic to the sphere, 78 not, 1 Complex plane (z-plane, Gauss plane, Argand-Gauss plane, A rgand-Cauchy or, 1 plane), 71 xor, 1 Complex projective n-space, 299 Connectivity, 196 Complex representation of Fourier series, 654 Connectivity number, 256, 257, 261, 412

785 Index

Connectivity pair, 262 Convex envelope, 252, 480 Conservative Hamiltonian system, 694 Convex function, 678, 679 Consistency, 3 Convex functional, 20 Constant, 581, 610 Convex hull, 250, 255, 278 Constant map, 29 Convex optimization problem, 684 Constant morphism, 599 Convex part, 252 Constant path, 279, 297 Convex programming, 670, 673 Constant return to scale, 758 Convex set, 26, 277, 674 Constant sheaf, 552 properties, 675 Constant symbols, 511 separation, 676 Constraint, 670, 671 Convex set and convex function, 679 Constraints, 681 Convex set spanned by X, 278 Constructibility by straightedge and compass, 130 Convexity, 196, 252, 674 Construction by straightedge and compass, 129 Convexity in optimization problems, 674 Construction of C, 69 Convolution, 487, 488, 495, 497, 662, 668 Construction of an n-sided polygon, 131 Convolution of distributions, 668 Construction of C, 69 Convolution operation, 460 Construction of N, 62 Coordinate ball, 542 Construction of Number System, xvii Coordinate chart, 269, 541, 542 Construction of Q, 64 Coordinate curve, 398 Construction of R, 66 Coordinate curve on a patch, 398 Construction of Z, 59, 64 Coordinate domain, 542 Construction problem, 129 Coordinate function, 544 Constructions by straightedge and compass, 162 Coordinate neighborhood, 542 Constructive set theory, 16 Coordinate neighborhoods, 315 Contact configurations, 394, 395 Coordinate neighbourhood, 544 Contiguous simplicial maps, 296 Coordinate neighbourhood (coordinate patch), 541 Continuation, 308 Coordinate patch, 542 Continuation of a path, 308 Coordinate plane, 157 Continuity, 325, 329, 423 Coordinate ring, 531, 532 global, 200 Coordinate transformation, 414, 542 lo cal, 200 Coordinates, 157 Continuity at a point, 50 in affine planes, 157 Continuity of a sequence, 204 in projection planes, 157 Continuity of complex functions, 423 Coordinates of a tensor, 406 Continuous bijection, 194 Coplanar, 164 Continuous deformation, 401, 469 Coplanar vector, 181 Continuous distribution, 620 Coplanarity, 181 Continuous extension of a function, 326 Coproduct, 299, 591, 603 Continuous function, 325 Coproduct (direct sum), 591 Continuous geometry, 193 Coproduct of objects, 310 Continuous invariant, 193, 195, 197 Coresolution, 558 Continuous linear form, 238 Corestriction, 30 Continuous linear functional, 21 Coretraction, 598 Continuous linear map, 237 Coriolis, 360 Continuous logarithm, 472 Correction of abnormal values, 630 Continuous map, 50, 193, 194, 200, 219 Correction of heteroscedasticity, 635, 636 Continuous multilinear map, 239 Correlation, 149, 156, 625 Continuously differentiable curves Correlation coefficient, 625 explicit representation, 358 Correlation dimension, 731 parametric representation, 359 Correlation integral, 731 Continuum, 35, 49 Correlation of a projective plane, 156 Continuum hypothesis, 15, 35, 49, 578 Correlation theory, 625 Contour, 428, 429 Correlogram, 643 Contour integral, 429, 430 Correspondence, 677 Contour integral , 430 Correspondence between differential complex and null sequence, 522, 534 Contour integration, 429 Correspondence between manifolds, 414 Contour winding number, 430 Cosecant, 175 Contracted curvature tensor, 409 Coset, 84, 526, 599 Contractibility, 249 C osine, 175 Contractible, 249, 275 C ospan, 601 Contractible loop, 546 Cotangent, 175 Contractible space, 275, 305 Cotriple, 593 Contractile arcwise connected space, 249 Countability axioms, 211, 212 Contractile space, 249 Countable, 3, 18, 211, 542 Contractility, 545 Countable basis, 211, 539 Contracting homotopy, 291, 303 Countable everywhere dense part, 236 Contracting homotopy of complex, 291 Countable local basis, 543 Contraction, 229, 696 Countable product, 217 Contraction mapping, 229 Countable set, 18, 34, 78, 211 Contraction of volume, 722 Countably compact set, 211 Contragedient matrix, 406 Countably infinite, 18, 34, 211 Contragredient representation, 406 Countably infinite = Denumerable, 18, 211 Contraposition, 54 Countably infinite set, 18 Contravariant, 408 Countably metacompact space, 211 Contravariant functor, 265, 273, 594 Countably paracompact space, 211 Contravariant Hom functor, 273 Counterclockwise, 175 Contravariant index, 407, 409 Counting, 37 Contravariant tensor, 408, 410 Counting number, 33, 62 Contravariant tensor algebra, 407 Counting of poles, 476 Contravariant valency, 408 Counting of zeros, 476 Contravariant vector, 410 Coupling of frequency, 722, 726, 734 Control, 670 Cousin problem, 481 Control of process, 685 C ovariance Cov(x, y), 625 Control theory, xviii, 669 Covariant, 408 Convention of summation, 405 Covariant derivative, 417, 418 Convergence Covariant functor, 265, 273, 594 of a sequence, 50 Covariant Hom functor, 273 in terms of balls, 50 Covariant index, 407, 409 sim ple, 234 Covariant left-exact functor, 557 uniform, 234 Covariant tensor, 406, 408, 410 Convergence criterion, 323, 340 Covariant tensor algebra, 407 Convergence in C, 423 Covariant tensor of order 1, 406 É Convergence in norm, 488 Covariant valency, 408 Convergence of a filter base, 205 Covariant vector, 409 Convergence of a sequence, 50, 234 Cover, 26, 197, 208, 296, 306, 415 Convergence of a sequence in terms of balls, 50 Cover of a simplicial complex, 296 Convergence of Fourier series, 487 Covering, 26, 193, 197, 208, 306, 415 A-P convergence, 487 Covering family, 568, 607 almost everywhere convergence, 488 Covering homotopy lemma, 306 convergence in norm, 488 Covering homotopy property, 562 weak ∗ convergence, 488 Covering homotopy theorem, 306 Convergent estimator, 626 Covering map, 574 Convergent infinite product, 324 Covering map (covering projection), 574 Convergent product, 324 Covering projection, 306, 574 Convergent sequence, 66, 67, 322 Covering sieve, 607, 608 Convergent sequence of functions, 327 Covering space, 204, 306, 574 Conversion formulas, 661 Covering space of a topological space X, 306 Convex, 196, 275 Covering transformation, 307 Convex combination, 675 Covering transformation (deck transformation), 307

786 Index

Cr -atlas, 414 Decimal point, 68 Cr -structure, 541 Deck transformation, 307 Cramer Formula, 107 Decomposition into irreducible elements, 141 Creation of a torus, 737 Decomposition into prime ideals, 142 Crisp set, 16 Decomposition into simple elements, 335 Criterion of identifiability of the model, 642 Decompostion into simple elements, 118 Critical (or rejection) region, 617 Dedekind, 65 Cross product, 13, 182 Dedekind ring, 142 Cross-cap, 412 Deduction theorem, 4 Cross-caps, 256 Deductive system, 3 Cross-sectional data, 624, 633 Definable set, 512 Cross-sectional model, 631, 634 Definiendum, 5 Crosscap, 300 D efi niens, 5 Crossword dictionary order, 518 Definite hermitian form, 136 CRS property, 207 Definite positive hermitian form, 242 Cube duplication, 130 Definition, 5 Cubic parabola, 395 Definition by comprehension, 11 Cumulative distribution, 620 Definition by recurrence, 5 Cumulative distribution function, 620, 621 Definition of an attractor, 733 Cumulative distribution function (c.d.f), 616 Definitions of chaos, 732 Cumulative hierarchy of sets, 15, 578 Deformation, 569, 573 Cup product, 318—320, 550 Deformation retract, 277, 573 C url, 316 Degeneracy conditions, 699, 702 Curl theorem, 348, 412 Degree, 117, 283, 293 Curry-Yorke model (C-Y), 741 Degree of a continuous map, 293 Curvature, 177, 393, 394, 398, 415, 418 Degree of a loop, 503 of Riemann, 415 Degree of a map, 472 Curvature formula, 395 Degree of a mapping, 413 Curvature line, 403 Degree of a polynomial, 109 Curvature of a curve, 419 Degree of a vertex, 260 Curvature of a plane curve, 397 Degree of an algebraic number, 117 Curvature of a regular patch, 401 Degree of an element in a graded ring, 318 Curvature of a Riemannian space, 419 Degree of antipodal map, 304 Curvature of a surface, 419 Degree of freedom, 617, 622 Curvature of circle, 394 Degrees of freedom, 629 Curvature of helix, 394 Del, 343, 347 Curvature vector, 393, 394 Del operator, 348 Curve, 214, 215, 392 Delta of Dirac, 668 of Jordan, 445 Demonstration, 5, 6 regular, 445 Demonstration by the absurd, 5 Curve length, 358 Dence trajectory, 722 Curve length in R3 (or R2), 358 Denjoy integral, 369 Curve torsion, 394 Denjoy theorem, 692 Curve with isolated point, 336 Dense, 24, 198, 211, 542, 702 C urves in R3, 392 Dense morphism, 572 Curvilinear abscissa, 359, 393 Dense orbit, 709 Curvilinear coordinates, 343, 398 Dense part, 221 Curvilinear integral, 360, 411, 445 of a topological space, 219 case of continuous paths, 468 Dense set, 65, 198 of a gradient, 361 Dense valuation, 145 of real functions, 361 Density of orbits (DO), 733 properties, 445 Density of rational function, 475 Cusp, 702, 736 Denumerability, 33—35 Cusp and Generalized Hopf bifurcations, 712 of the set of rationals, 34 Cusp bifurcation, 712, 735 Denumerable, 3, 18, 211 Cusp bifurcation (Bogdanov, Takens), 712 Denumerable set, 18, 34, 53, 54, 78, 211 Cusp singularity, 702 Dependence of criteria (population), 623 CUSUM statistic, 631 Dependent variable, 617, 685 CUSUM test (CUmulative SUM), 631 D epth, 525 CUSUM-SQ statistic, 631 Derivability, 329 C ut, 65 Derivable function, 329 Cut point, 262 Derivate, 329 CW complex, 298, 300 Derivation, 114 CW complexes, 298, 300 Derivation of distributions, 496, 668 CW decomposition, 300, 535 Derivation operator, 526 CW space, 300, 535 Derivation operator (differentiation), 526 CW subcomplex, 300 Derivation rule, 329 CW-complex, 535 Derivation rules, 392 CW-complexes, 196 Derivative, 119 Cycle, 86, 260, 285, 474, 534 Derivative of a rational fraction, 119 Cyclic decomposition, 136 Derivative of cumulative distribution function, 620 Cyclic decomposition theorem, 136 Derived category, 592 Cyclic Galois group, 128 Derived functor, 556, 557, 597 Cyclic group, 87, 257 Derived polynomial, 114, 121 Cyclic multiplicative group, 114 Derived set, 369 Cyclic object, 303 Derived structure, 38 Cyclic order, 160 Desargues and Pappus-Pascal theorems, 155 Cyclic order of equivalence classes, 151 Desargues theorem, 155 Cyclic order relation, 160 Descartes, 4 Cyclic permutation, 91 Descartes folium, 335 Cyclic subbasis, 136 Descending chain condition (DCC), 523 Cyclic subgroup, 73 Descriptive geometry, 150, 173, 174 Cyclic subgroup of T, 73 Destination, 598 Cyclic subgroup of unit circle, 73 Destruction of a torus, 739 Cyclic subspace, 136 Detection of error autocorrelation, 633 Cyclic variable, 695 causes, 633 Cyclic vector, 136 detection, 633 Cyclically ordered, 160 Breusch-Godfrey test, 634 Cyclically reduced word, 519 Durbin-Watson test, 634 Cycloid, 722 Detection of heteroscedasticity, 635 Cyclotomic equation, 131 Detection tests of multicollinearity, 632 Cyclotomic polynomial, 116, 125 Determinant, 105, 133, 717 Cyclotomic polynomial of order k, 125 calculation rules, 106 Determinant calculation rules, 106, 344 δ-functor, 557 Determination d-stable, 753 of argument, 455 d-stable matrix, 757 of logarithm, 455 Damped oscillation, 663 Determination of a measure, 350 Darboux-Ribaucour thriedron, 401 Determination of lag number, 646 Daughter wavelet, 665 Determination of lag numbers, 640 De l’Hospital rules, 334 Deterministic walk, 644 De Moivre number, 73 Devaney, 732 De Rham cohomology, 316, 364, 412, 449, 501 Developable surface, 404 Decimal comma, 68 Deviation, 230, 685 Decimal expansion, 34, 67 Dfferential form periodic, 68 change of variable, 410 of rational number, 68 Diagonal approximation, 319 terminating, 68 Diagonal morphism, 565 Decimal numeral system, 56 Diagonalization, 133, 134 Decimal numeral system (base ten, base-10, denary), 56 Diagram, 273, 601

787 Index

Diagram of the pure rotation and its lift, 692 Direct limit (inductive limit), 601 Dice, 259, 615 Direct limit (of algebraic systems), 536 Dickey-Fuller Direct limit (of sets), 535 augmented, 644 Direct product, 266, 375, 519, 524, 602 Dickey-Fuller test (DF), 644 Direct product of modules, 102 Diffeomorphism, 542, 546, 686, 691 Direct product of subgroups, 89 hyperbolic linear, 692 Direct product of subgroups (commutative group), 89 Diffeomorphism and periodic points, 692 Direct similarity, 165, 452 Diffeomorphism of the circle, 687 Direct sum, 133, 266, 290, 318, 376, 519, 591, 602 Diffeomorphisms and flows, 686 Direct system, 536, 559, 601 Difference, 8, 11 Direct system in a category, 601 Difference filter order, 644 Directed angle, 75 Differentiability, 329, 342 Directed graph, 259, 582, 600 complex, 424, 549 Directed graph (digraph), 600 real, 424, 549 Directed graph (or digraph), 259 Differentiable 2-chain, 506 Directed line segment, 150 Differentiable atlas, 414, 415, 542 Directed order, 205, 579 Differentiable concave function, 678 Directed order (filtrant order), 579 Differentiable convex function, 678 Directed preorder, 535 Differentiable function, 329 Directed segment, 150 Differentiable manifold, 414, 415, 513, 540, 697 Directed set, 23, 205, 535, 579 Differentiable operator, 372 Directed straight-line, 150 Differentiable p-form, 410 Directedness, 259 Differentiable structure, 540, 541 Direction curves, 404 Differential, 331, 410, 555 Directional blowing-up, 702 Differential (coboundary map), 555 Directional derivate, 342 Differential 1-form, 443 Directions of principal curvature, 403 Differential 2-form, 449 Directly congruent, 160 Differential algebra, 526 Directrix, 185, 186 Differential calculus, xvii, 328, 341 Dirichlet box principle, 53 Differential calculus in Rn, 341 Dirichlet conditions, 655 Differential complex, 522, 534 Dirichlet Pigeonhole principle, 53 Differential Equation, 377 Dirichlet problem, 487, 492 Differential equation, xvii Disappearance of saddle-node separatrix, 738 (linear) homogeneous, 382 Disconnected space, 202, 203 exact, 379, 380 Disconnectivity, 203 explicit form, 378 Discontinuity, 194, 326 implicit, 381 Discrete category, 582 initial conditions, 378 Discrete complex Fourier coefficients, 656 initial-value problem, 378 Discrete complex Fourier transformation, 656 of second order, 381 Discrete distance, 220 order, 378 Discrete dynamical system, 686, 715, 723 structurally stable (robustness), 723 invariant manifolds, 723 with one variable, 378 Hadamard-Perron theorem, 723 with separated variables, 379 separatrix surfaces, 723 Differential equation system, 383 transverse homoclinic points, 723 Differential field, 526 periodic orbits, 723 Differential form, 316, 409, 410 properties of ω-limit set, 723 closed form, 448 topological conjugacy of discrete systems, 723 exact form, 448 types of equilibrium points (Steady states), 723 in the plane, 448 Discrete Haar wavelet transformation, 666 on a differentiable manifold, 410 Discrete Laplace transformation, 664 on an open subset of C, 498 Discrete random variable, 620 Differential form of degee 1, 443 Discrete space, 226 Differential forms of degree 2, 449 Discrete topology, 198 Differential geometry, xvii, 392, 415 Discrete valuation, 145 curves, 392 Discrete wavelet transformation (DWT), 666 Differential graded algebra, 527 Disjoint sets, 26 Differential k-forms, 138 D isk, 71 Differential manifold, 686 Disk with holes, 450, 452 Differential module, 522, 523, 527 Displacement, 158, 186 Differential operator, 374, 415, 668 Dissipative baker map, 732 Differential p-form, 316 Dissipative system Differential ring, 526 transitions stemming from linear stability loss, 718 Differential sheaf, 553 Distance, 66, 76, 184, 219, 220 Differential topology, 415 between two parts of a metric space, 220 Differentially stable (d-stable), 753 Hamming, 220 Differentially stable matrix, 757 Hausdorff, 220 Differentiation, 708 Minkowski, 220 Differentiation of degree n, 289 on a normed vector space, 220 Diffusion model, 637, 638 p-adic, 220 Digraph, 600 sem i, 220 Dilation, 165 ultra-metric, 220 Dilation parameter, 666 Distance (in a normed vector space), 237 Dimension, 294 Distance function, 727 Dimension axiom, 287, 302, 312, 549, 553, 597 Distance on a space of functions, 217 Dimension axiom theorem, 312, 317 Distinction between εt and et, 628 Dimension invariance, 212 Distinguished subgroup, 88 Dimension of a simplicial complex, 294 Distinguished triangle, 591 Dimension of the measure µ, 731 Distributed lag models, 639 Dimension theory, 212 Distributed parameter system, 739 Dimensions, 730 Distribution, 391, 496, 659, 668, 726 Dimensions through invariant measures, 731 leptokurtic, 644 correlation dimension, 731 norm al, 644 dimension of a measure, 731 Distribution of lags generalized dimension, 731 fi nite, 640 information dimension, 731 infinite, 640 Lyapunov dimension, 731 Distribution of order 0, 496 Local Hausdorff dim. (Douady-Oesterlé), 731 Distribution of order 1, 496 metric dimensions Distribution of zeros, 463 Box-counting dimension or capacity, 730 Distribution theory, 654, 668 fractals, 730 Distributions, 616 Haussdorff dimension, 730 Distributions and subharmonic functions, 492 self-similarity, 731 Distributive lattice, 23 Dimetric projection, 174 Distributive law, 42, 516 Dini theorem, 234 Distributivity, 8, 9 Diophantine equation, 144 Divergence, 316, 347, 348 Dirac, 668, 727 Divergence formula, 489 D irac δ function and distribution, 659 Divergence of a vector field, 347 Dirac mass, 496 Divergence theorem, 348, 364, 412 Dirac measure, 725 Dividend, 64 Direct demonstration, 5 D iv ides, 95 Direct family, 535 Divisibility, 55, 141 Direct family (of algebraic systems), 536 a divides b, 141 Direct family (of sets), 535 a is a divisor of b, 141 Direct homothety, 165 a is a factor of b, 141 Direct isometries in R3, 164 b is a multiple of a, 141 Direct isometry, 160 b is divisible by a, 141 Direct limit, 266, 536, 601, 602 in a field, 145 Direct limit (filtered colimit), 591 Divisibility in N, 55

788 Index

Divisibility in Z, 55 Embedding, 66, 416, 520, 544, 594 Divisibility in polynomial ring, 120 of metric planes in projective planes, 157 Divisibility theory, 145 Embedding (imbedding), 544 Divisible, 141 Embedding of Q in R, 66 Divisible group, 317 Embedding space, 416 Divisor, 64, 141, 146, 147 Embedding Whitney theorem, 416 Divisor function, 435 Empirical coeffcicient, 625 Domain, 14, 29, 30, 38, 93, 422, 502, 512, 519 Empirical estimator of the variance, 627 Domain of definition, 29, 38, 119, 519 Empirical t-student, 625 Domain of variation, 581 Empty map, 30 Dominant coefficient, 109 E m pty set, 7 Dominant morphism, 572 End of a path, 279 Dominant term, 109 End of edge of simplicial complex, 297 Dominant term of a polynomial, 109 Endoarrow, 600 Dot product, 21, 182 Endofunctor, 593 Douady-Oesterlé theorem, 731 Endogenous variable, 625, 626 Douady-Oesterlé theorem (for diff. eq.), 731 Endomorphism, 33, 134, 520, 598 Double cone, 311 Endomorphism (of algebraic system), 520 Double integral, 355, 356 Endomorphism eigenvector, 189 Double value principle, 1 Endomorphism reduction, 134 Double-angle formula, 177 E ngle, 648 Doubling the cube, 75 Engle-Granger, 649 Doubly periodic function, 436, 437 Engle-Granger algorithm, 648 Doubly stochastic, 616 E nough -injectives, 555 DS process, 644 Enough injectiveH objects, 555 with drift, 644 Enough injectives, 555—557 without drift, 644 Enough projectives, 556 DS process (random walk model), 644 Enriched category, 588 DS processes (Difference Stationary), 644 Enriched category identity, 588 D ual Enriched functor, 597 algebraic, 238 Entering variable, 669 topological, 238 Entire function, 433, 457 Dual basis, 238, 405 Entire ring, 43, 93, 516 Dual category, 583, 584 Entire series, 132, 328, 428, 453, 457 Dual space, 19, 21, 139, 370, 405, 408 convergence sets, 440 Dual space functor, 273 of several variables, 440 Dual space of V (V ∗= Hom(V, k)), 408 with nonnull radius, 333 Dual universal coefficients theorem, 318 Entire series expansion, 421, 426, 457 Dual vector space, 105, 405 of rational function, 426 Duality principle, 671 Entire series expansion of circular functions, 337 Duffing attractor, 709, 710 Entourages, 230 Duffing system, 710 E ntropies Dummy index, 39 metric entropy, 727 Dummy variable, 630 topological entropy, 727 Dupin indicatrix, 403, 419, 420 Entropy, 727 Durbin-Watson test, 639 Kolmogorov-Sinai, 727 Durbin-Watson test (DW), 634 m etric, 727 Dyadic wavelet transformation, 666 Enumerable, 18, 211 Dynamical model, 648 Enumeration, 37 Dynamical system, xviii, 684, 686, 714 E nvelop e conservative, 715 convex, 480 dissipative, 715 upp er, 230 volume-contracting (or dissipative), 715 Envelope of holomorphy, 480 volume-preserving (or conservative), 715 Epigraph, 349 E-cyclic object, 303 Epimenides, 13 ε-neighborhood, 199 Epimorphism, 38, 520, 598, 599 Eccentricity, 185 Epimorphism (epi), 264 Edge, 258 Epimorphism (epic), 598 Edge connectivity, 262 Epimorphism (or epi), 598, 606 Edge number, 263 Equal sets, 7, 14 Edge of simplicial complex, 297 Equality of two pairs, 27 Edge path, 297 Equality sign, 3 Edge path group, 297 Equality test of variances, 636 Edge-preserving bijection, 258 Equalizer, 305, 604 Eigen decomposition, 133 Equation, 106 Eigendirection, 189 Equation of 3rd degree Eigenfunction method, 390 resolution, 128 Eigenspace Equation of 4th degree stable, 708 resolution, 129 unstable, 708 Equation of a straight line, 157 Eigensubspace, 133 Equation of line, 183 Eigenvalue, 133, 385, 722 Equation of second degree, 186 Eigenvalue analysis of the matrix eTR, 717 in two coordinates, 186 Eigenvalue problem, 133, 134 Equation of tangent plane, 399 Eigenvector, 385 Equation of the first degree, 157 Eilenberg, 575 Equation solvable by radicals, 129 Eilenberg criterion, 474 Equation with determinant, 184 Eilenberg-Mac Lane, 576 Equation with point and normal, 184 Eilenberg-Moore algebras, 594 Equation with vector product, 184 Eilenberg-Moore algebras (of a monad), 594 Equicontinuity, 235 Eilenberg-Moore category, 594 Equicontinuous, 235 Eilenberg-Steenrod axioms, 549, 553 family of maps, 235 Eilenberg-Steenrod theorem, 302 Equicontinuous family, 235 Eilenberg-Steenrod theorem extension, 302 Equicontinuous family of maps, 235 Eilenberg-Zilber theorem, 305 Equidecomposable, 17, 18 Einheit, 42, 515 Equilateral hyperbolic function, 178 Einstein summation, 343, 405 Equilibrium in a distribution economy, 677 Einstein summation convention, 405, 407 Equilibrium point of type (m,k), 719 Eisenstein series, 435 Equipollent, 29, 33 Elastic transformation, 193 Equipollent bipoints, 161 Elation, 156 Equipotence, 33, 34, 36 Elem entary i-chain, 504 Equipotent, 29, 33 Elementary symmetric function, 113 Equipotent set, 34 Elementary topos, 608 Equivalence, 594 Ellipse, 185 Equivalence class, 28, 201, 276 Ellipse circumference, 358 Equivalence in a category, 273 Elliptic, 158 Equivalence of categories, 266, 594, 596 Elliptic function, 436 Equivalence of flows, 692 Elliptic function order, 436 Equivalence of loops, 445 Elliptic geometry, 179 Equivalence of paths, 445 Elliptic invariant, 435 Equivalence relation, 18, 28, 64, 67, 201, 276, 415 Elliptic metric plane, 154 Equivalent absolute values, 145 Elliptic plane, 152 Equivalent arcs, 414 Elliptic point, 402, 403, 419 Equivalent covering spaces, 307, 309 Elliptic right triangle, 180 Equivalent generating system, 200 Elliptic umbilic, 403 Equivalent local families, 711 Embed a metric plane in a projective plane, 157 Equivalent norms, 237 Embedded hypersurface, 544 Equivalent norms (in a normed vector space), 237 Em b edded m-dimensional tori, 720 Equivalent representations of Fourier integrals, 661 Embedded submanifold, 544 Equivalents sets (equipotent sets, 47

789 Index

Equivariant map, 740 Evaluation map, 309 Ergodic dynamical system, 725 Even and odd polynomials, 109 Ergodic measure, 725 Even polynomial, 109 Ergodic theory, 725 Evenly covered open set, 306 E rror, 626 Event, 615, 618 Error correction model (ECM), 648 Evolute, 397 E rror εt, 626 Evolute of a curve, 396, 397 Error of approximation, 332 Evolute of a plane curve, 398 Error of specification, 626 Evolute of an arc, 397 Essential extension, 555 Evolvent, 396 Essential monomorphism, 555 Exact differential equation, 379, 380 Essential singularity, 421, 464 Exact differential form, 380 Essential submodule, 555 Exact form, 448 Essential supremum, 391 Exact functor, 595 Essentially surjective, 594, 595 Exact homology sequence, 291 Essentially unique, 57 Exact sequence, 290, 313, 509, 554, 605 Estimation method of d Exact sequence (acyclic complex), 290 aggregate variance, 652 Exact triangle, 291 aggregated absolute value, 652 Exact triangle theorem, 291 periodogram, 652 Exactness axiom, 302, 553 R /S, 652 Exactness of a sequence, 290 variance of the residuals, 652 Exchange rules, 3 wavelets, 652 Excision, 292 Whittle, 652 Excision 1, 292 Estimation methods of nonlinear models, 638 Excision 2, 292 Estimation of parameters, 626 Excision axiom, 302, 553, 597 Estimation procedures for error correlation, 634 Excision corollary, 301 Estimator, 626 Excision theorem, 296, 317 BLUE, 629, 634, 635 Excluded third principle, 2 of fractional integration parameter, 652 Exclusion restrictions, 642 of Hurst exponent, 652 Exhaustion sequence, 458 Estimator of coefficients, 626 Exhaustion sequence of compact sets, 458 Estimator of Generalized Least Squares, 633 Existence of covering space, 308 Estimator of the variance of the error, 627 Existence of null element, 9 Et(´ X), 568, 569 Existential quantifier, 10 Et´ /X, 569 Exogenous variable, 625 Etalé space, 610 Expansion, 696 E tale, 568 Expansion of 1/Γ, 468 Etale cohomology, 568, 570, 571 Expansion of 1/ζ, 467 Etale cohomology group, 570 Expectation, 616, 621 Etale cover, 567, 571, 610 Expectation (of a random variable), 616 Etale fundamental group, 568 Expectation of the error, 626 Etale map, 568 Expected value, 616 Etale map of schemes, 610 Experiment, 616 Etale morphism, 568, 569 Explained variability (SSE), 627 Etale neighborhood, 568 Explanatory variable, 617, 626 Etale presheaf, 610 Explosion, 746 Etale presheaf on a scheme, 610, 611 Exponent, 68 Etale sheaf, 610 irrational, 68 Etale site, 568 rational, 68 Etale site of a scheme, 610, 611 Exponential, 72, 336 Etale topology, 568 Exponential absolute value, 145 Etale topology (algebraic geometry), 568 Exponential Fourier transformation, 661 Etale topos, 568, 610 Exponential function, 72, 336, 454 Etale X-scheme, 568 Exponential functor, 608 Euclid algorithm, 58, 111 Exponential law, 309 Euclid axioms, 149 Exponential object, 580, 586 Euclid lemma, 56, 57, 60, 95 Exponential series of matrices, 384 Euclid lemma (for polynomials), 111 E xpression Euclid lemma in Z, 60 identical, 2 Euclid theorem, 56, 169 satisfiable, 2 Euclidean algorithm, 142 Ext functor, 558 Euclidean distance, 220 Extended complex plane, 421, 422 Extended complex plane (C, dˆ), 422 Euclidean division, 55, 59, 109 É Euclidean division algorithm, 110 Extended complex plane C, 421 É Euclidean division of polynomials, 109 Extended Euclid algorithm, 112 Euclidean function, 142 Extended real line, 223, 422 monotone, 142 compact, 223 Euclidean geometry, 149, 150, 160 Extended real number line, 217 Euclidean group E(n), 163 Extending by linearity, 285 Euclidean metric, 153, 193, 199, 409 Extension, 573 Euclidean moves, 163 of absolute values, 146 Euclidean plane, 154 of natural numbers, 64 Euclidean polynomial division, 120 of semiring of natural numbers, 63 Euclidean ring, 142, 143 Extension field, 65, 69, 122, 531 Euclidean space, 211 Extension field degree, 122 Euclidean transformation, 161 Extension field of Q, 65 Euclidean/non-Euclidean metric plane, 153 Extension field of R, 65, 69, 422 Euler affinity, 169 Extension of a field, 122 Euler approximation of Duffing attractor, 709 Extension of a map, 29, 573 Euler characteristic, 296, 413 Extension of a semigroup, 91 Euler characteristic (of a genus), 412 Extension of first-order predicate calculus, 3 Euler characteristic (of a topological space), 412 Extension of Q, 67 Euler constant, 338 Extension of Q to R, 65 Euler criterion, 144 Extension of R to C, 65 Euler differential equation, 373 Extension theorem, 228 Euler formula, 76, 296 Extensional definition, 11 Euler formulas, 74, 654 Exterior, 214 Euler function, 143, 338 Exterior algebra, 137—139, 318 Euler line, 166, 167 of a vector space, 139 Euler product formula, 58 of alternating multilinear forms, 139 Euler theorem, 96, 167, 403 Exterior algebra (Grassmann algebra), 137 Euler totient, 57, 96 Exterior angle, 162 Euler totient function, 57, 96 Exterior angle of a polygon, 162 Euler’s displacement theorem, 186 Exterior calculus, 138 Euler-Cauchy method, 387 Exterior derivative, 316, 364, 450 Euler-Lagrange theorem, 83, 84 Exterior derivative of a 1-form, 450 Euler-Poincaré characteristic, 296, 413 Exterior differential, 410 Euler-Poincaré characteristic of (X,E), 301 Exterior differential form, 138 Euler-Poincaré characteristic of a polyhedron, 296 Exterior differentiation, 410 Euler-Poincaré characteristic of K, 296 Exterior differentiation of differential forms, 138 Eulerian chain, 260 Exterior normal, 489 Eulerian cycle, 260 Exterior point, 195, 198 Eulerian function, 651 Exterior power, 316 Eulerian graph, 260 Exterior product, 138—140, 497 Eulerian multiplier, 379 of two antisymmetric multilinear forms, 138 Eulerian path, 260 universal properties, 140 Eulerian trail, 260 Exterior product (wedge product), 139 Eulerian walk, 260 External center of similitude, 166 Euclidean division in Z, 59 External composition law, 43, 516

790 Index

External direct sum, 524 Filtration, 301 External homothetic center, 166 Final topology, 201 Extracted family, 31 Finitary operation, 510, 511 Extracted sequence, 221 Finite Abelian group, 89 Extraction of a root nth, 77 Finite distribution of lags, 640 Extraordinary cohomology theory, 302 Finite extension, 122 Extrema, 373 Finite field, 125, 126 Extrema under constraints, 346 Finite Fourier transformation, 657 Extreme value theorem, 209 Finite Galois extension, 127 Extremum, 209 properties, 127 Extrinsic curvature, 418 Finite Galois extension of Q, 131 Finite graph, 259 F-acyclic, 557 Finite group, 82 F-acyclic object, 556 Finite Laplace transformation, 656 F-algebra, 594 Finite morphism, 566 F-distribution, 617 Finite presentation, 569 F-homomorphism, 521 Finite product, 217 F-model set, 303 Finite rank operator, 375 F-projective n-space, 299 Finite separable field extension, 569 F-test, 617 Finite separable k(y)-scheme, 569 Face, 294, 303 Finite set, 5, 53 Factor, 141 Finite simple extension, 123 Factor ring, 43, 526 Finite support, 40, 108 Factor through, 571, 586 Finite type, 318, 566 Factorable polynomial, 120 Finitely based chain complex, 297 Factorial ring, 124, 142 Finitely generated, 295 Factorization, 57 Finitely generated ideal, 529 Factorization theorem, 57 Finitely generated module, 566 Faithful functor, 266, 594 Finitely generated morphism, 566 Faithfully flat module, 565 Finitely presented, 297 False proposition, 1 Finitely presented algebra, 529 Fam ily, 31 Finitely presented R-algebra, 529 Family canonically associated with a set, 31 Finsler manifold, 540 Family of elements, 31 First axiom of countability, 538, 543 Family of parts of a set, 32 First category, 232 Family of polynomials with graded degrees, 109 First countable, 211, 542 Family of sets, 13, 31 First countable basis, 211 Fano axiom, 158 First countable space, 211 Fano axiom (P5),(P5’), 158 First Cousin problem, 481 Farrar-Glauber test, 632 First de Rham cohomology vector space, 501 Fast Fourier transformation (FFT), 656 First difference model, 635 Fast wavelet transformation (FWT), 666 First fundamental form, 400 Fatou theorem, 488 of a sphere, 400 Fb er, 268 patch theory, 400 Fber bundle, 268, 560 First homology group, 475 Feasible polyhedron, 671 First quasi-differences, 634 Feasible region, 669, 670 First-countable, 538 Feasible region of optimization problem, 671 Feasible value, 671 First-countable space, 211, 212 Feasible value of optimization problem, 671 First-difference filter, 644 Feedback effect, 647 First-order differential equation, 378 Feigenbaum attractor, 715 First-order predicate calculus, 2 Fermat conjecture, 96 First-order predicate calculus with identity, 3 Fermat equation, 144 Fisher law, 627 Fermat infinite descent, 52 Five lemma, 291 Fermat last theorem, 96 Fixed line, 173 Fermat little theorem, 96 Fixed point, 30, 86, 173, 687, 730 Fermat method of infinite descent, 52 Fixed point of a projective collineation, 173 Fermat number, 131, 148 Fixed point theorem, 229 Fermat principle of infinite descent, 52 Fixed-point theorems, 677 Fermat theorem, 93, 96 Fixed-point theorems (Brouwer and Kakutani), 674 Ferrari method, 129 Flabby sheaf, 559 Feuerbach circle, 167 Flag, 150, 159 Fiber, 53, 275, 306, 520, 560, 569 Flasque resolution (flabby resolution), 559 Fiber (algebraic geometry), 560 Flasque sheaf, 559 Fiber (topology), 560 Flasque sheaf (flabby sheaf), 559 Fiber bundle, 562 Flat manifold, 418 Fiber of a map, 53, 560 Flat module, 565 Fiber of cover, 306 Flat morphism, 565, 569 Fiber product, 314, 604 Flat point, 419 Fiber space, 268, 562 Flat unramified morphism, 569 Fibonacci numbers, 741 Flo or, 61 Fibration, 268, 314, 562 Floor function, 61, 624 Fibration (Hurewicz fiber space), 314 Floquet characteristic multiplier, 717, 746 Fibre bundle, 562 Floquet matrix, 717, 722, 734 Fibre product, 265, 266 Floquet theorem, 699, 716 Fibre square, 266 Floquet theory, 717, 734 FIEGARCH, 652 Floquet theory in dissipative systems, 717 Field, 38, 43, 81, 92, 98, 143, 360, 516 Flow, 714, 715 left, 79 Flow of a differential equation, 714 noncommutative, 79 Flow on the torus, 693 Field adjunction , 122 Flow-box, 691 Field characteristic, 125, 126 Flow-box theorem, 691 Field extension, 122 Flows and differential equations, 689 Field homomorphism, 99, 521 Focal parameter, 185 Field line, 348 Focus, 185, 186 Field of complex numbers, 422 Folding map, 310 algebraically closed, 77 Folium , 335 Field of complex numbers is algebraically closed, 115 Force, 360 Field of fractions, 99, 117 Force along a curve, 361 Field of fractions (fraction field, field of quotients, quotient field), 117 Forced pendulum, 747 Field of fractions of a ring, 98 Forecast interval, 631 Field of fractions of R, 98 Forecast reliability, 631 Field of parallel vectors, 417 Forecasting error, 627 Field of quotients, 99, 100, 117 Forest, 260, 261 Field of rational numbers, 64 Forgetful functor, 273, 597 Field of residue classes, 145 Formal proof, 4 Field theory, 81, 514 Formal system, 4 FIGARCH (fractionally integrated GARCH), 652 Formally étale, 569, 570 FIGARCH (p,d,q), 652 Formally étale morphisms of schemes, 570 Filter, 19, 205, 208, 222, 223, 543 Formally étale=Formally smooth+Formally unramified, 570 C auchy, 231 Formally smooth, 570 determined by a sequence, 231 Formally unramified, 570 Fréchet, 222 Formula of a parallelogram, 242 of neighborhoods, 222 Formula of the median, 242 smallest, 222 Fortuitous correlation, 625 Filter base, 22, 204, 205, 209, 543 Forward regression, 633 Filter bases, 222 Foundations of mathematics, 1 Filtered colimit, 591 Four-color problem, 261 Filtered set, 23, 535 Fourier, 389 Filtrant category, 589 Fourier coefficient, 442, 457, 460, 493, 653

791 Index

Fourier coefficients, 244, 656 totient, 68 determination for symmetric functions, 655 Function from R to Rm, 341 symmetry of the first kind, 655 Function from Rn to R, 342 symmetry of the fourth kind, 655 Function from Rn to Rm, 344 symmetry of the second kind, 655 Function from Rn to Rn, 344 symmetry of the third kind, 655 Function Γ, 459 determination with numerical methods, 655 Function ζ, 458 Fourier cosine transformation, 657, 661 Function of differentiable real variable, 329 Fourier expansion, 654 Function ℘, 434, 437 Fourier integral, 477, 661 Function preserving structure, 576 Fourier integral in complex representation, 661 Function space, 233, 309 Fourier series, 435, 436, 487, 653, 654 Function symbols, 511 complex representation, 654 Function theory, xvii, 421 important properties, 654 Function theory (complex analysis), xvii Fourier series and Fourier integrals, 655 Functional, 19, 20, 370, 391 Fourier series and transform Functional analysis, xvii, 370 usual writings, 654 Functional axiom, 14 Fourier series convergence, 653 Functions, 29 Fourier series expansion, 457 Functions of several variables, 440 complex, 436 Functor, 256, 257, 264, 265, 272, 273, 576, 594 real, 436 Functor categories, 576 three different forms, 655 Functor category, 586 Fourier series integration, 653 Functor Hn, 271 Fourier sine transformation, 657, 661 Functor in algebraic topology, 257 Fourier sum, 654 Functor Spec, 572 Fourier transform, 390, 653, 654 Functorial, 587 continuous writing, 654 Functorial isomorphism, 587 discrete writing, 654 Functorial method, 256 three continuous writings, 654 Functorial variable, 3 Fourier transformable, 661 Functoriality, 587 Fourier transformation, 495, 657, 661 Fundamental (homotopy) group, 250 Fourier transformation (definition), 661 Fundamental group, 247, 248, 250, 255, 280 Fourier-Plancherel transformation, 654 of a connected polyhedron, 254 Fppf cover, 567 Fundamental groupoid, 280 Fpqc cover, 567 Fundamental groups of polyhedra, 297 Fréchet filter, 222, 543 Fundamental homotopy groups , 256 Frénet basis, 394, 397, 398 Fundamental matrix, 385 Frénet formulas, 394, 397 complex, 385 Frénet trihedral, 394 real, 385 Fractal dimensional, 730 Fundamental neighborhood, 306 Fractals, 730 Fundamental neighborhood of base space, 306 Fraction field, 99, 117 Fundamental relations Fractional Brownian motion, 651 normal, geodesic curvature, curvature, 401 Fractional Gaussian process, 651 Fundamental relations between χn,χg,χ, 401 Fractional ideal, 146 Fundamental sequence, 66 Fractional integration parameter, 651 Fundamental set, 8 Fram es, 666 Fundamental structure, 37, 38 Frechet differential, 372 Fundamental system, 383 Fredholm alternative, 377, 390 Fundamental system of neighborhoods, 200, 221, 230, 234 Fredholm determinant, 390 Fundamental tensor, 407, 418 Fredholm integral equation, 390 Fundamental theorem Fredholm operator, 376 for homogeneous linear diff. equations, 716 Fredholm theorem, 374, 390 for non-homogeneous linear equations, 716 Fredholm theorems, 390 theory of patches, 405 Fredholm theory, 390 Fundamental theorem of algebra, 77, 283, 284 Free A-module, 316 Fundamental theorem of arithmetic, 57 Free abelian, 295 Fundamental theorem of arithmetic (for polynomials), 111 Free Abelian group, 518, 519 Fundamental theorem of curve theory, 395 Free algebra, 518, 523 Fundamental theorem of integral calculus, 349, 352 Free associative algebra , 523 Fundamental theorem of projective geometry, 156 Free chain complex, 303 Fundamental theorem of Riemannian geom., 418 Free commutative algebra, 518, 523 Fuzzy set theory, 16 Free functor with basis in , 303 Fuzzy set:, 16 M Free group, 103, 254, 297, 298, 502, 518, 519 G-equivariant, 307 Free group of rank m, 298 G-isomorphism, 308 Free group with basis, 297 G -m ap, 307 Free homotopic loops, 502 G-module homomorphism, 521 Free homotopy, 274 G-paradoxical set, 8 Free ideal ring, 523 G -set, 306 Free ideal ring (fir), 523 G-space, 306, 562 Free mobility, 159 G ödel, 3, 37 Free module, 102, 524 Gödel completeness theorem, 3 Free module (homological algebra), 524 Gödel incompleteness theorem, 3 Free monoid, 518, 523 Gödel second incompleteness theorem, 3 Free object, 519, 580 Gabor transformation, 666 Free object (universal algebra), 519 Galerkin method, 749 Free orthogonal system, 243 Galois field, 43, 126, 516 Free part, 102 Galois group, 126 Free product, 297 Galois group of a polynomial, 128 Free R-module, 518, 523 Galois map, 126 Free resolution, 556 Galois theory, 38, 81, 126, 514 Free ring, 524 Galton board, 621 Free variable, 2 Galtonian curve, 621 Free Z-module, 519 Gambling, 615 Frege, 4, 14 Gamma function, 337, 651 Frequency locking, 741, 744 GARCH(p,q), 652 Frequency spectrum, 662 GARMA process, 651 Frequency table, 620 Gateaux differential, 372 Freudenthal suspension theorem, 315 Gauss and Mainardi-Codazzi relations, 405 Frobenius method, 417 Gauss complex plane, 76 Frobenius theorem, 417 Gauss curvature, 403 From Torus to chaos, 739 Gauss formula, 501 Front face, 318 Gauss formulas, 404 Frontier, 294 Gauss function, 622 Frontier point, 195, 198, 219 Gauss lemma, 144 Frontier set, 195 Gauss lemma (for polynomials), 111 Fubini formula, 54 Gauss plane, 422 Fulcrum, 190 Gauss relations, 405 Full functor, 594 Gauss theorem, 57, 60, 348 Full subcategory, 264, 281, 593 Gauss theorem in Z, 60 Fully faithful functor , 594 Gauss trihedron, 404 Function, 14, 29 Gauss-Newton algorithm, 638 harmonic, 425 Gaussian curvature, 403, 419, 420 Lagrange, 673 Gaussian curvature of a surface, 420 logarithmic barrier, 674 Gaussian elimination, 133 one-to-one, 29 Gaussian white noise, 643 penalty, 673 Gavrilov-Shilnikov, 748, 749 subharmonic G C H , 15 nontrivial, 491 Gegenbauer, 651 test, 668 Gegenbauer frequency, 652

792 Index

General equation of degree n, 129 Global Spec, 571 General equilibrium, 677 Global stability, 759 General equilibrium (economics), 677 Globally stable equilibrium, 757 General form of dynamic model, 684 GLS estimator, 634, 635 General form of state-space model, 685 Gluing, 274, 547, 606, 610 General Lie group, 132 Gluing (concatenation, collation), 606 General linear group, 131, 172, 416 Gluing axiom, 551, 606, 607 General linear group GLn(q), 172, 191, 417 Gluing condition, 551, 606 General linear group GLn(R), 172, 417 Gluing lemma, 274 General linear model, 628 Gnomonic projection, 179 General monic polynomial, 128 G o de, 559 General orthogonal group GOn(q), 191 Godement resolution, 559 General orthogonal group GOn(q, F ), 172, 191 Golbach conjecture, 1 General position, 278 Goldbach conjecture, 148 General quantifier, 2, 10 Golden rule, 758 General set theory, 15 Golden theorem (Gauss), 144 General term, 240 Goldfeld-Quandt test, 636 General topology, 193 Gollub-Benson, 726 General valency, 408 Gompertz model, 638 Generalization of Cauchy criterion, 228 Goodwin cyclical growth model, 761 Generalization of VECM, 649 Goodwin model, 753 Generalized ARFIMA process, 651 G rade, 525 Generalized continuum hypothesis (GCH), 15 Graded algebra, 263, 476, 523 Generalized derivative, 391 Graded co-ring, 320 Generalized dimension, 731 Graded coring, 320 Generalized element, 581 Graded module, 263, 522, 523 Generalized Euclidean space, 21, 137 Graded ring, 264, 270, 318, 476, 522, 523 Generalized function, 391, 659 Graded sheaf, 553 Generalized function (or distribution), 391 Graded valuations, 110 Generalized functions, 20 Gradiant, 489 Generalized Hopf bifurcation, 712, 713, 736 Gradient, 316, 342, 343 Generalized Hopf singularity, 713 Gradient operator, 412 Generalized integral, 354, 355 Gradient theorem, 343, 412 Generalized Lagrangian, 681 Graeco-Latin square, 613 Generalized lagrangian, 679 Graeffe method, 341 Generalized least squares (GLS), 633 G ram m ar, 5 Generalized sequence, 205, 206, 322 Granger, 647, 651 Generating family, 89 Granger representation theorem, 649 Generating part, 102 Graph, 258, 299, 560, 600 Generating set, 519 Graph automorphism, 259 Generating system, 199 Graph characteristic polynomial, 134 Generator, 84, 297, 519, 524, 581, 591, 609 Graph coloring, 261 Generator , 95 Graph coordinates, 539 Generator of an ideal, 95 Graph cycle (circuit), 260 Generator of an ideal , 111 Graph edge, 258 Generator part, 168 Graph isomorphim, 258 Generators of an abelian group, 285 Graph of a correspondence, 27 Generatrix, 400 Graph of a map, 27, 30 Generatrix lines, 404 Graph of binary relation on a space, 299 Generic element, 581, 610 Graph of map, 29, 519 Generic property, 724 Graph representation, 258 Generic property (or typical property), 724 Graph theory, xvii, 258 Gentzen-type, 2 Graph vertices, 258 G enus, 256 Graph without loop and spindle, 259 Genus theorem, 145 Graphical integration, 355 Geodesic, 179, 372, 373, 401, 417 Graphical projection, 174 Geodesic curvature, 401 anamorphic projection, 174 Geodesic line, 177, 401 auxiliary view, 174 Geometric arc, 392 parallel projection, 174 Geometric complex, 251, 255 axonometric projection, 174 Geometric congruence, 152 axonometric projection (dimetric), 174 Geometric construction, 129 axonometric projection (isometric), 174 Geometric fiber, 560, 569 axonometric projection (trimetric), 174 Geometric point, 560 multiviews, 174 Geometric realization, 295 multiviews (elevation), 174 Geometric sequence, 322 multiviews (floor plane), 174 Geometric series, 323 multiviews (section), 174 Geometry, xvii, 149 oblique projection, 174 absolute, 149, 151 oblique projection (cabinet), 174 analytic, 149, 181 oblique projection (Cavalier), 174 axiom C (connection), 154 orthographic projection, 174 axiom D (completion) Dedekin, 160 perspective projection, 174 axiom H (hyperbolic), 154 Grassman manifold, 138 axiom P (polar trilateral), 154 Grassmann algebra, 137 axiom R (rectangle), 153 Grassmann coordinates, 138, 140 axiom s A,A∗ (bisector), 159 Grassmannian, 138 axioms A1,A2,A3 (affine), 155 Gray code, 667 axioms M1,M2,M3 (metric plane), 151 Great circle, 179 axioms P1,P2,P3 (projective), 155 Greatest common denominator (GCD), 69 axioms P5,P5’ (Fano), 158 Greatest common divisor, 57 descriptive, 150, 173, 174 Greatest common divisor (gcd.), 120, 143 elliptic, 149, 179 Greatest common divisor in Z, 60 Euclidean, 150 Greatest element, 9, 45, 46 hyperblolic, 176 Greatest lower bound, 9, 46 hyperbolic, 149, 152 Green formula, 489, 501 Lobachevsky, 152 Green theorem, 285, 316, 348, 364 Lobachevsky-Bolyai-Gauss, 149, 152 Green-Riemann formula, 363, 411 non-Euclidean, 149 Green-Riemann theorem, 363 ordinary, 150 Grelling, 13 parabolic, 149 G rid, 171 projective, 149 Grid search procedure (Hildreth-Lu), 635 spherical, 179 Gridding, 171 Gereno lemniscate, 336 Grobman-Hartman theorem, 723 G erm , 551 Grobman-Hartman theorem (for discrete sys.), 723 Germ of an analytic set, 560 Gronwall lemma, 754 Giraud theorem, 609 Gross substitutability assumption, 760 Glesjer test, 636 Grothendieck, 269, 575 G lide, 162 Grothendieck abelian category, 591 Glide reflection, 162 Grothendieck cohomology, 550 Global bifurcation, 736, 737, 746 Grothendieck pretopology, 607 Global bifurcations, 734 Grothendieck theorem, 567, 570 Global continuity, 200 Grothendieck topology, 569, 607 Global element, 581, 610 Grothendieck topology axioms, 607 Global element (global section, point), 581, 610 Grothendieck topos, 608 Global homoclinic bifurcations, 738 Grothendieck trace formula, 571 Global maximum, 679 Grothendieck universe, 16, 577 Global mean value, 491 Grothendieck universe U, 577 Global minimum, 679 Group, 38, 42, 81, 82, 515 Global Proj, 572 Abelian, 42, 516, 517 Global section, 558, 563, 581, 610 isometry, 149, 151

793 Index

K lein, 42 Heaviside unit step function, 658 of rigid motions, 151 H eight, 166 similtude, 149 H eine, 223 transformation, 149 Heine theorem, 223, 224 Group by arrows, 590 Helix, 393, 396 Group commutator , 519 Henon map, 715 Group generators, 103, 254 Hereditarily finite set, 578 G roup GLn(K) of nonsingular matrices, 106, 173 Hereditary property, 35 Group homomorphism, 38, 84, 521 Hereditary set, 14, 16, 578 Group isomorphism, 85 Hermitian conjugate operator, 375 Group kernel, 88 Hermitian form, 136, 137, 242 Group law, 82 Hermitian inner product, 21, 136, 375 Group object, 310 Hermitian inner product (inner product), 375 Group of affine collineations G, 172 Hermitian integral operator, 377 Group of automorphisms, 38 Hermitian norm, 137, 374, 377 Group of complex numbers of modulus 1, 72 Hermitian operator, 375 Group of displacements, 159 Hermitian scalar product, 21, 137 Group of finite type, 89 Hermitian space, 21, 137 Group of invertible elements of a ring, 93 Hermitian symmetric sesquilinear form, 136 Group of isometries, 187 Hermitian symmetry, 136 Group of n-boundaries, 290 left semilinear, 136 Group of n-cycles, 290 right linear, 136 Group of nonsingular matrices, 106 Hesse form, 184 Group of orthogonal collineations, 173 Hessenberg, 155 Group of periods of ω, 503 Hessian, 346 Group of regular matrices, 106 Hessian matrix, 346 Group of relative n-boundaries, 291 Heteroclinic point, 703 Group of relative n-cycles, 291 Heteroclinic saddle connection, 747 Group of rigid motions, 151 Heterological adjective, 13 Group of rotations, 41, 515 Heteroscedastic, 644 Group order, 42 Heteroscedasticity, 633, 635 Group theory, 81, 82, 514 Heteroscedasticity), 643 Groupoid, 280, 297, 582 Hexadecimal numeral system, 56 Groupoid category, 594 Hexagon axiom, 587 G roup oid π(K), 297 Hexagon identities, 587 Groupoid theorem, 280 Higher aithmetic, 51 Groups obtained by extension, 92 Higher aithmetic (number theory), 51 Guckenheimer, 732 Higher Dimensional Algebra (HDA), 594 Guckenheimer-Holmes, 699, 713 Higher order partial derivative, 344 Gysin sequence, 320 Higher-dimensional categories, 576 Higher-order predicate calculus, 3 h-Cobordism, 548 Hilbert basis, 244 H-group, 311 Hilbert norm, 137 -injective, 555 Hilbert plane, 151 Hh-level, 312 Hilbert problem, 3 H-space, 283 Hilbert program, 4 Hénon map, 696, 715 Hilbert space, 21, 137, 199, 211, 241, 242, 371, 489 Hénon quadratic map, 696 Hilbert space of finite dimension, 244 Haar wavelet, 666 Hilbert-type, 2 Hadamar formula, 453 Hildreth-Lu method, 635 Hadamard matrices, 667 Histogram, 615 Hadamard three circles theorem, 491 Hjelmslev, 151 Hadamard-Cartan & Gromoll-Meyer theorem, 420 Hjelmslev theorem of perpendiculars, 152, 153 Hadamard-Perron theorem, 720 Hjemslev, 157 Hadamard-Perron theorem (for discrete sys.), 723 Hodge star, 412 Hahn-Banach separation theorem, 26 Hodge star operator, 412 Hahn-Banach theorem, 24, 45, 236, 238 Holes, 506, 549 Hahn-Banach theorem (geometric form), 26 Holomorphic C-C-function, 424, 548, 549 Hairy ball theorem, 294 Holomorphic continuation, 426 Half-angle formulas, 177 Holomorphic function, 333, 421, 425, 426, 453, 548 Half-ball, 230 local behavior, 477 Half-cone, 184 on an open set of S , 482 Half-exact functor, 595 2 Holomorphic function in an annulus, 464 Half-period ratio, 435 Holomorphically convex, 480 Ham-sandwich theorem, 320 Holomorphically convex set, 480 Hamilton rules, 79 Holomorphy, 421, 424, 440 Hamiltonian, 694, 695, 716 equivalent formulations, 421 Hamiltonian chain, 260 extension of the concept, 422 Hamiltonian circuit, 260 Holomorphy of C-C-functions, 422 Hamiltonian cycle, 260 Holomorphy of C-C-functions, 422 Hamiltonian flow, 694 É Hamiltonian function, 694 Hom functor, 595 Hamiltonian graph, 260 Hom-set, 582, 583 Hamiltonian path, 260 Homeomorphic, 194 Hamiltonian system, 694, 711, 724 Homeomorphic image, 197 Hamiltonian system with n-degrees of freedom, 694 Homeomorphic spaces, 219 Hamming distance Homeomorphic topological space, 200 between two finite sets, 220 Homeomorphism, 193, 194, 219, 248, 271, 538, 539, 546, 687, 733 n-tuplets, 220 Homeomorphism (topology), 538 Handle, 256, 300, 432 Homeomorphism onto its image, 545 Hankel transformation, 657 Homoclinic and heteroclinic orbits, 721 Hardy space, 460, 489 Homoclinic loop, 747 Harmonic, 726 attracting, 747 Harmonic analysis, 655 repelling , 747 Harmonic division, 166, 168 Homoclinic point, 702, 703 Harmonic function, 425, 485 Homoclinic points, 709 Harmonic homology, 156, 158 Homoclinic saddle connection, 710, 747 Harmonic majorant, 491 Homoclinic saddle connection theorem, 747 Harmonic series, 323 Homoclinic tangle, 697, 709 Harmonic upper-bound, 491 Homogeneity, 20 Harmonicity and holomorphy, 485 Homogeneous coordinate, 532 Harnack inequalities, 486 Homogeneous coordinates, 252 Hartman-Grobman, 697 Homogeneous element, 528 Hartman-Grobman theorem (for diffeom.), 697 Homogeneous function, 533 Hartman-Grobman theorem (for flow), 697 Homogeneous ideal, 270, 528, 533 nd Hartogs, 441 Homogeneous integral equation of 2 kind, 374 Hartogs theorem, 494 Homogeneous medium, 661 Hasse diagram, 9, 45 Homogeneous polynomial, 533 Hasse Diagram (order diagram), 45 Homogeneous system, 107 Hasse principle, 79 Homogeneous transformation, 20 Hausdorff axiom, 219 Homographic equation, 184 Hausdorff dimension of the measure µ, 731 Homographic transformation, 439 Hausdorff dimension properties, 730 classification, 439 Hausdorff distance, 220 composition, 439 Hausdorff paradox, 17 Homography, 483 Hausdorff quotient space, 299 Homological algebra, 263 Hausdorff separation axiom, 206 Homological equation, 698, 700 Hausdorff space, 193, 199, 206, 207, 538, 539 Homologous cycles, 474 Haussdorf dimension, 730 Homologous points, 166 Heat equation, 389, 661 Homology, xvii, 156, 251, 255, 263, 549 Heaviside, 668 Homology class, 287, 290

794 Index

Homology class of zn, 287 Hyperbolic axiom (H), 154 Homology classes, 285 Hyperbolic decreasing, 651, 652 Homology functor, 286, 597 Hyperbolic equilibrium point of type (m,k), 723 Homology group, 251, 255, 257, 263, 287, 475, 550, 749 Hyperbolic fixed point, 697, 747 Homology group (Betti group), 257 Hyperbolic function, 177, 337 Homology of spheres, 292 hyperbolic cosecant, 177 Homology sequence, 597 hyperbolic cosine, 177 Homology theory, 251, 255—257, 263, 302, 476, 509 hyperbolic cotangent, 177 Homology theory (H, ∂), 302 hyperbolic secant, 177 Homomorphic image, 510, 521 hyperbolic sine, 177 Homomorphism, 38, 81, 84, 514, 521, 598, 606 hyperbolic tangent, 177 Homomorphism (abstract algebra), 521 Hyperbolic geometry, 176 Homomorphism (morphism), 598 Hyperbolic invariant set, 708 Homomorphism as cartesian product, 521 Hyperbolic linear diffeomorphism, 692, 696, 697, 705 Homomorphism of Banach algebras, 494 Hyperbolic linear flow, 696 Homomorphism of complexes, 521 Hyperbolic metric plane, 154 Homomorphism theorem, 88 Hyperbolic model, 152 applications, 88 Hyperbolic non-wandering set, 709 Homomorphism theorem for modules, 102 Hyperbolic nonlinear fixed point, 697 Homomorphism theorem for rings, 100 Hyperbolic periodic orbit, 697 Homomorphisms of an algebraic system, 590 Hyperbolic plane, 152 Homoscedasticity, 628 Hyperbolic point, 402, 403, 419 Homothetic circles, 166 Hyperbolic Riemann surface, 415 Homothetic figures, 165 Hyperbolic saddle-point, 710 Homothetic ratio, 165 Hyperbolic segment, 176 Homothetic triangle, 166 Hyperbolic set, 708 Homothety, 165 Hyperbolic singular point (for flow), 697 Homothety and point at infinity, 165 Hyperbolic singularity, 699 Homothety-reflection, 166 Hyperbolic structure, 708 Homothety-rotation, 165 Hyperbolic trygonometry, 178 Homotopic, 247, 749 Hyperbolicity, 702 Homotopic continuous maps, 472 Hypercomplex number, 35 Homotopic maps in a basic sense, 249 Hyperparallel, 152, 154 Homotopic path edges, 297 Hyperplane, 717 Homotopic paths, 247, 502 Hypersurface, 717 Homotopic paths with fixed endpoints, 247, 470 Hypertetrahedron, 251, 255 Homotopic topological space, 248 Hypervolume, 251, 255 Homotopy, xvii, 214, 247, 248, 254, 270, 274, 279, 468, 469, 472, 562, 573 Hypoellipticity, 492 of continuous maps, 248 Hypograph, 349 of polygonal chains, 254 Hypotenuse, 175, 178 Homotopy (fundamental group), 502 Hypothesis, 2, 5 Homotopy axiom, 288, 302, 553, 597 Hypothesis H 0, 625 Homotopy axiom for pairs, 292 Hypothesis of normality of errors, 626 Homotopy axiom theorem, 288, 317 Hypothesis of recurrence, 5, 39 Homotopy category, 274, 305, 311, 592 Hypothesis testing, 616, 617, 622 Homotopy category (hTop), 311 Hysteresis, 748, 751 Homotopy class, 248, 274 I-adic topology, 533 Homotopy class (mapping class), 573 i-boundaries, 497 Homotopy equivalence, 248, 249, 274, 305 i-boundary, 506 Homotopy equivalence in homotopy category, 274 i-cocycles, 501 Homotopy extension property, 314 i-chains, 497, 504 Homotopy extension theorem, 301 i-chains with integral coefficients, 504 Homotopy function, 248 i-coboundaries, 501 Homotopy group, 247, 250, 309, 312 i-cycle, 90, 506 Homotopy identity, 283 I-depth, 525 Homotopy invariance, 284 i-faces, 251, 255 Homotopy invariant, 573 i-th étale cohomology group, 571 Homotopy lifting property, 314, 573 i-th cohomology group, 571 Homotopy of continuous maps, 248 i-th homology group, 556 Homotopy of paths, 247 i-th right derived functor, 557 Homotopy sequence of a fibration, 314 Ideal, 22, 43, 93, 100, 142, 269, 516, 523, 528 Homotopy sequence of a pair, 314 Ideal element, 4, 149 Homotopy sequence of a weak fibration, 315 Ideal line, 157 Homotopy set, 573 ideal object, 4 Homotopy theory, 214, 270 Ideal plane, 157 Homset, 582, 585 Ideal plane of a metric plane, 157 Hooke law, 377 Ideal point, 154, 157 Hopf algebra, 320 Ideal sheaf, 552 Hopf bifurcation, 711, 718, 735 Ideal sheaf (algebraic geometry), 552 generalized, 713 Ideal theory, 142 subcritical, 712 Idempotence, 8 supercritical, 712 Idempotent, 93 Hopf bifurcation (normal form), 733 Idempotent element, 33 Hopf bifurcation theorem, 712 Idempotent ring, 10 Hopf fibration, 315 Identically independently distributed (i.i.d.), 643 Hopf recurrence theorem, 718 Identification, 275, 299 Hopf singularity, 711 Identification space, 574 Hopf singularity of type (p, +) or (p, ), 713 − Identities for complex arguments, 177 Hopf singularity of type p, 713 Identity, 3, 576 Hopf theorem, 320 Identity element, 42, 43, 515, 516 Hopf-Landau model, 739 Identity functor, 273 Horizontal composition, 264, 585 Identity map, 29, 30, 173, 219 Horner schema, 341 Identity matrix, 105 Horner scheme, 109 Identity object, 586 Horseshoe, 708 if and only if (iff), 5 Horseshoe diffeomorphism, 703, 705, 708 IGARCH (integrated GARCH), 652 canonical example, 705 Image, 14, 29, 30, 85, 290, 519, 520 dynamics on symbol sequences, 707 Image of homomorphism, 85 symbolic dynamics for horseshoe diffeo., 707 Image of morphism, 85 Horseshoe map, 731 Imaginary part, 76 Hosking, 651 Imaginary part , 70 HSP theorem, 514 Imaginary part of z, 70 hTop, 311 Immeasurable frequencies, 726 Hurewicz fiber space, 314 Immersed submanifold, 544 Hurewicz fiber space (fibration), 314 Immersion, 213, 546 Hurewicz homomorphism, 315 Immersion (algebraic geometry), 571 Hurewicz map, 289 Immersion (algebraic topology), 546 Hurewicz theorem, 288, 289, 315 Immersion theorem, 213 Hurewicz theorem (1), 289 Impact multplier, 647 Hurewicz theorem (2), 315 Implicit differential equation, 381 Hurewicz theorem (absolute version), 315 Implicit differentiation, 335 Hurewicz theorem (relative version), 315 Implicit function, 345 Hurst exponent (H), 651 Important properties of Fourier series, 654 Hurst exponent estimator, 652 asymptotic behavior, 655 Hurst statistic, 651 convergence of function in mean, Parseval equation, 654 Hurwitz theorem, 477 Dirichlet conditions, 655 Hypebolic convergence, 650 least mean squares error of a function, 654 Hyperbola, 185, 186 Improper point, 155 Hyperbolic, 158 Impulse function (Dirac δ function), 659 Hyperbolic attractor, 748 In bijection with, 36, 47

795 Index

Inaccessible cardinal, 15, 578, 579 Integral calculus, xvii, 348 Inadequacy of Zariski topology, 567 Integral closure, 146 Incidence, 150 Integral domain, 43, 93, 99, 100, 120, 142, 457, 516, 523 Incidence-preserving transformation, 157 divisibility, 141 Incident, 152 Integral domain (entire ring), 43, 93, 516 Inclusion, 11 Integral domain by prime ideal, 100 Inclusion function, 520, 591 Integral equation, 374 Inclusion functor, 584 (homogeneous) of the second kind, 374 Inclusion map, 520, 544, 599 (nonhomogeneous) of the second kind, 374 Incomplete basis theorem, 103 classification, 374 Indefinite integral, 349, 352, 353 of 1st kind, 374 of algebraic functions, 354 Integral equation of the first kind, 374 of rational functions, 353 Integral equations of the second kind, 374 of rparticular functions, 354 Integral function, 352, 457 Independent events, 619 Integral kernel, 377 Indeterminates, 120 Integral mean-value, 351 Index, 430, 471 Integral of a 1-form on a part of R, 445 Index change formula, 40 Integral of a differential 2-form, 501 Index gymnastics, 407 Integral of a differential form, 410 Index of a cycle, 474 according to a parameter, 411 Index of a loop, 473, 504 Integral of closed 1-form on a continuous path, 469 Index of loop with respect to a point, 470 Integral of surface, 362 Indicator, 57 Integral on a continuous path, 469 Indicator variable, 630 Integral on a piecewise- 1 path, 445 Indicator variable (or dummy variable), 630 Integral on an orientedC curve, 447 Indirect demonstration, 5 Integral operator, 374 3 Indirect isometries in R , 164 Integral property, 93, 98 Indirect least squares (ILS), 642 Integral scheme, 564 Indirectly conformal mapping, 438 Integral transformation, 656 Individual, 2 Integral transformations, 656 Induced homomorphism, 545 finite Fourier transformation, 657 Induced law, 32 finite Laplace transformation, 656 Induced map, 273, 554 for functions of several variables, 657 Induced order, 38, 45 Fourier cosine transformation, 657 Induced orientation, 286 Fourier sine transformation, 657 Induced topology, 194, 199, 201, 538, 543 Fourier transformation, 657, 661 Induction, 35 Hankel transformation of order, 657 Induction axiom, 4, 50, 62 Laplace transformation, 656, 657 Induction hypothesis, 5, 39 Laplace-Carson transformation, 657 Inductive definition, 5 Mellin transformation, 657 Inductive hypothesis, 5 of functions of one variable, 656 Inductive limit, 550, 552, 601 one-sided Fourier transformation, 657 Inductive limit (direct limit), 601 Stieltjes transformation, 657 Inductive order, 46 two-sided transformation, 656 Inductive ordered set, 47 Walsh functions, 667 Inductive set, 47 wavelet transformation, 665 Inertia degree, 146 Z-transformation, 664 Infi mum , 46 Integrand, 349 Infinite cyclic group, 296 Integrating factor, 379 Infinite descent, 52 Integration by parts, 353 Infinite distribution of lags, 640 Integration methods, 353 Infinite graph, 259 Integration of 0-form, 506 Infinite product, 324, 467 Integration of 1-forms, 506 Infinite product expansion, 467 Integration of a C-C-function, 425 Infinite set, 34 Integration of series, 354 Infinite-dimensional F-projective space, 299 Integration rule, 351 Infinitesimal calculus, xvii Integration rules, 425 Infix form, 32 Integration theorems, 362 Inflection point, 334 Integrations term by term, 354 Information dimension, 731 Interior, 195, 214, 216, 333, 536, 537, 548 Initial condition problem, 386 Interior angle, 162 Initial conditions, 378 Interior angle of a polygon, 162 Initial morphism, 580 Interior point, 195, 198, 219, 536, 537, 674 Initial object, 310, 580 Interior point methods, 674 Initial property, 580 Interior product, 139 Initial segment, 48, 65 Interleave sequence, 216 Initial topology, 201 Intermediate field, 122 Initial-value problem, 378 Intermediate value theorem, 203, 226, 326 Injection, 310, 520 Intermittency, 738, 745 Injective arcs on manifold, 414 type 1, 718, 746 Injective envelope (injective hull), 555 typ e 2, 746 Injective homomorphism, 520, 555 typ e 3, 746 Injective homomorphism (monomorphism), 520 Internal (composition) laws of lattice, 9 Injective immersion, 544, 545 Internal angle bisector, 167 Injective linear transformation, 546 Internal center of similitude, 166 Injective map, 29, 31, 519 Internal composition law, 39, 41, 42, 515 Injective module, 554 Internal direct sum, 525 Injective object, 555, 556 Internal Hom functor, 586 Injective patch, 406 Internal homothetic center, 166 Injective resolution, 555, 556 Internally homologous, 373 Inner product, 21, 136, 137, 182, 375 Interpolation, 339 Inner product space, 136, 137 Interpolation theory, 339 Inner-product space, 21 Interpretation, 513 Innovation, 647 Interpretation (logic), 513 Input variable, 684 Interpretation (model theory), 513 Inscribed angle, 163 Interpretation of multipliers, 681 Inscribed angles in a circle, 163 Intersection, 8, 9, 11, 13 Inscribed circle, 176 Intersection of a simple arc with a sphere, 396 Insertion function, 591 Intersection of neighborhoods, 195 Inset, 707, 709 Intersection of open sets, 195 Insets and invariant manifold, 709 Intersection of subcomplexes, 290 Instability, 717 Intput variable, 685 subharmonic, 734 Intrinsic curvature, 418 Instrumental variable, 637 Intrinsic geometry, 400, 401, 415 Integer, 64 Intrinsic parameter, 393 algebraic structure, 64 Intrinsic property, 197 order structure, 64 Intuitionism, 4 Integer modulo n, 87 Intuitionistic logic, 4 Integer part, 61 Intuitionistic propositional logic, 4 Integer part of a rational fraction, 118 Intuitionistic Zermelo-Fraenkel (IZF), 16 Integers Invariance of dimension, 294 calculation rules, 64 Invariance of domain, 294 Integrability of the logarithm, 463 Invariance of integral by homotopy, 469 Integrable system, 695 Invariant Cantor set, 692 Integral, 146 Invariant manifold, 697, 720, 747 Integral calculation Invariant manifold theorem, 697 complement formula (meromorphic embedding of ζ), 477 Invariant manifold theorem (for diffeomorph.), 697 Fourier integral, 477 Invariant manifold theorem (for flow), 697 of a rational function without real poles, 477 Invariant measure, 19, 718, 724, 727 trigonometric inttegral, 477 Invariant set, 690, 708

796 Index

Invariant set stability, 715 Jarque-Bera test, 643 Invariant sets, 715 Jensen formula, 463, 492 Invariant subgroup, 84, 599 Johnson theorem, 494 Inverse, 32, 40 Joint entropy, 727 Inverse edge path, 297 Jordan arc, 214, 215, 392, 393 Inverse element, 32, 42, 82, 86, 515 admissible parameterization, 393 Inverse function, 327, 546 Jordan areolar measure, 365 Inverse function theorem, 390 countable union of parts of R366 Inverse homeomorphism, 219 of rectangles (possibly degenerated), 365 Inverse image, 29, 30, 498, 519, 520, 560 problem, 366 Inverse image (preimage), 520 Jordan basis, 135 Inverse image of a 1-form by a map, 444 Jordan block, 135 Inverse image of a 2-form by a map, 450 Jordan canonical form, 136 Inverse Laplace transformation, 657 Jordan canonical form theorem, 136 Inverse map, 30 Jordan Curve, 255 Inverse mapping theorem, 568 Jordan curve, 197, 214, 392, 445 Inverse matrix, 106 closed, 197 Inverse on the left, 32 Jordan curve in C, 423 É Inverse on the right, 32 Jordan curve theorem, 214, 294, 504 Inverse path, 279, 445 Jordan domain, 485 Inverse relation, 29 Jordan form, 698 Inverse symbol, 30 Jordan Lemma, 429 Inverse transformation, 666 Jordan loop, 444 Invertibility, 141, 344 Jordan matrix, 135 Invertibility and continuity, 327 Jordan matrix decomposition, 135 Invertible, 40, 93, 141 Jordan path, 444 Invertible element, 40, 141 Jordan regular curve, 445 in residue-class ring, 143 Jordan theorem, 473, 474 Invertible on the left, 32 Jordan-Brouwer separation theorem, 294 Invertible on the right, 32 Jordan-Brouwer theorem, 214 Invertible operator, 372 Joule, 360 Involute, 396 Joyeux, 651 Involute of a curve, 396 jth Betti number, 502 Involute of a plane curve, 398 Judge, 632, 634, 637 Involute of an arc, 396 Involution, 40, 134, 161, 240, 562 k(y)-scheme, 569 Involutive function, 134, 240 k-algebra, 568 Involutive map, 134, 240 k-connected graph, 261, 262 Involutive mapping, 134, 240 k-face, 279 Involutive transformation, 161 k-jet, 701 Inward perturbation, 747 k-simplex, 251, 255 Irrational number, 68 k-skeleton, 300 Irrational rotation number K-theory, 269, 562 (quasiperiodic behavior), 744 K-vector subspace, 108 Kähler differential, 527 Irrationality of √2, 65 Königsberg bridges, 258 Irreducibility criteria, 121 Künneth formula, 305 Irreducibility of polynomials, 121 Künneth formula for cohomology, 318 Irreducible, 141 Künneth theorem, 305 Irreducible affine algebraic set, 531 Kakutani, 674, 677 Irreducible element of a ring, 95 Kakutani fixed point theorem, 677 Irreducible form of a rational number, 62 KAM theorem, 724 Irreducible fraction, 68 Kelley-Morse set theory (KM), 15 Irreducible function, 530 Kelvin-Stokes theorem, 364 Irreducible polynomial, 110, 120, 121, 530 Kepler (approximation) method, 355 Irreducible prime element, 142 Kepler folium, 336 Irreducible scheme, 564 Kernel, 85, 88, 133, 161, 290, 557, 599, 605 Irrelevant ideal, 528 Kernel (in abstract algebra), 521 is divisor of, 44 Kernel for vector space, 521 is multiple of, 44 Kernel is a normal subgroup, 88 Isoclinal line, 379 Kernel of derivation function of polynomials, 114 Isocline, 379 Kernel of group homomorphism, 521 Isogonal mapping, 438 Kernel of homomorphism, 85 Isolated logarithmic singularity, 433 Kernel of homomorphism of differential complexes, 522 Isolated point, 195, 198 Kernel of linear transformation, 521 Isolated singular point, 428 Kernel of magma homomorphism, 521 Isolated singular solution, 381 Kernel of morphism, 85 asymptotic point, 381 Kernel of ring homomorphism, 521 centre, 381 Kernel pair, 588, 599, 605 nodal point (node), 381 K et, 409 saddle point, 381 Keynesian assumptions, 760 Isolated singularities on k-sheets, 433 K leene, 1 Isolated singularity, 464 K lein, 163 Isolated zeros, 462 Klein bottle, 255, 412 Isometric patches, 400 Klein group, 42 Isometric projection, 174 Klein model, 149, 152, 176 Isometric regular patches, 400 Klein test, 632 Isometry, 18, 151, 173, 193 K not, 197 canonical transformation, 187 Knot without crossing, 197 glide, 149 Kolmogoroff axioms, 618 identity, 149 Kolmogorov-Arnold-Moser theorem, 724 reflection, 149 Kolmogorov-Sinai entropy, 727 rotation, 149 Kolmogorov-Smirnov statistic, 631 translation, 149 Kondrakov theorem, 391 Isometry group, 149, 151, 173, 187 Korch curve, 731 Isometry group Is(E), 187 + Koyck model (geometric progression), 640 subgroup of displacements Is (E), 187 Koyck transformation, 641 subgroup of displacements Is−(E), 187 Kripke-Platek set theory (KP), 15 Isomorphic, 10, 606 Kronecker, 4 Isomorphic directed graphs, 600 Kronecker delta, 343 Isomorphic functors (naturally equivalent functors), 302 Kronecker formula, 413 Isomorphic graphs, 259 Krull dimension, 526 Isomorphic groups, 85 Krull topology, 533 Isomorphism, 33, 38, 81, 302, 514, 520, 598, 606 kth root of a real, 69 Isomorphism of groups, 85 Kuhn-Tucker conditions, 679 Isomorphism of ordered sets, 47 quantification, 682 Isomorphism theorem, 89 Kuhn-Tucker multipliers Isomorphism theorem for modules, 102 interpretation, 682 Isomorphism-dense subcategory, 595 uniqueness, 682 Isoperimeter, 373 Kuhn-Tucker theorem, 682 Isotone map, 66 Kupka-Smale, 709 Isotropic line, 158 Kuratowski graph, 263 Isotropic vector, 137 Kuratowski theorem, 263 Isotropy group, 483 Kuratowski-Pontryagin theorem, 263 Isotropy subgroup, 306 Kurtosis test, 643 ith face map, 286 2 ith k-simplex, 550 L -space, 392 L’Huilier theorem, 180 -shaped diagram, 601 L-adic étale cohomology, 570 JacobianJ matrix, 343, 344, 716 L-adic cohomology, 569, 570 Jacobson radical, 529 L-automorphism, 126

797 Index

L-globally stable equilibrium, 757, 758 Legendre polynomials, 339 α-limit point, ω-limit point, 690 Legendre symbol, 144 α-lim it set, ω-limit set, 715 Leibniz criterion of convergence, 324 L-locally stable equilibrium, 757 Leibniz formula, 114 L-stability, 757 Leibniz notation, 330 L∞[a, b] space, 371 Leibniz rule, 526 Löwenheim-Skolem theorem, 2 L ength, 447 Lagged variable, 634 of a curve, 447 Lagrange, 673 of a Jordan curve, 448 Lagrange equation, 750 of a path, 447 Lagrange form of Taylor remainder, 332 Length measure, 415 Lagrange formula, 395 Length measure on a regular patch, 400 Lagrange interpolating polynomial, 112 Lens space, 301 Lagrange interpolation, 339 Leontief input-output model, 669 Lagrange interpolation polynomial, 112 Leptokurtic distribution, 644 Lagrange method, 339 Leray theorem, 296 Lagrange multiplier, 346, 634 less than or equal to, 36, 44 Lagrange multiplier (LM), 634 Level homotopy, 312 Lagrange multiplier test, 630 Level of degeneracy (or codimension), 699 Lagrange polynomial, 112 Level set, 347, 520 Lagrange stability, 718 L ever, 190 Lagrange theorem, 83, 86 Levi-Civita, 407 Lagrangian theorem, 681 Levi-Civita connection, 418 Lagrangien, 681 Lexicographic product, 49 L amb ek, 577 Li-Yorke, 732 Laminar phase, 738 Lie algebra, 132 Laminarization, 746 Lie algebra of a Lie group, 132 Laplace transformation, 656, 657 Lie bracket, 132, 698 convergence, 657 Lie group, 131, 417 inverse Laplace transformation, 657 Lift, 268, 573, 688, 691 rules for the evaluation , 657 Lift (algebraic topology), 573 transform of special functions, 658 Lift (category theory), 573 Laplace-Carson transformation, 657 Lift (lifting), 573 Laplacian vector, 347 Lifted map, 740 Large category, 568, 582, 585 Lifting, 282, 573 Lateral classe, 84 Lifting criterion, 307 Latin square, 613 Lifting criterion theorem, 307 Lattice, 9, 22 Lifting lemma, 282, 306 Lattice ideal, 23 Lifting of maps, 216 Lattice maximal ideal, 23 Lifting property, 569, 570, 573 Lattice non-trivial ideal, 23 Limit, 220, 305, 602 Lattice of periods of ω, 503 Limit and cluster points of function with respect to a filter, 223 Lattice of power set, 10 Limit and colimit, 305 Lattice prime ideal, 23 Limit cycle, 690, 713, 717, 718, 720, 724, 730, 734 Lattice proper ideal, 23 Limit of a filter, 223 Lattice theory, 9 Limit of a map, 221 Lattices of sets, 9 Limit of a sequence, 221 Laurent series, 428, 464, 664 Limit of function, 325 Laurent series expansion, 428 Limit ordinal, 15, 48, 578 Law of composition, 29, 30 Limit point, 26, 195 Law of double negation, 2 Limit theorem, 620 Law of excluded middle, 4 Limit value of function, 325 Law of large numbers, 620 Lindelöf space, 212 Law of monotony, 63 L ine, 163 Law of non-contradiction, 2 Line angles, 75 Laws of Morgan, 2 Line at infinity, 157, 173 Lcal element, 581, 610 Line bundle, 561 Leading coefficient, 109 Line connectivity, 262 Leading coefficient of a polynomial, 109 Line integral, 343, 412, 429 Leading monomial, 518 Line integral of a vector field, 343, 412 Leading term, 109 Line of curvature, 403 Leading term of a polynomial, 109 Linear algebra, 132, 136 Least common multiple, 58, 121 affine structure, 133 Least common multiple (lcm.), 143 diagonalization and decomposition, 133 Least common multiple in Z, 60 Euclidean and Hermitian structure, 133 Least element, 45, 46 linear structure, 133 Least mean squares error of a function, 654 multi-linearity, 133 Least squares method Linear closure, 102 geometric interpretation, 628, 631 Linear combinations, 102 Least upper bound, 9, 45, 46, 160 Linear correlation coefficient, 625 Least upper bound theorem, 66 Linear differential equation, 380, 716 Least-squares, 339 autonomous, 716 Least-squares method, 339 homogeneous, 716 Leaving variable, 670 non-homogeneous, 716 Lebesgue differentiation theorem, 488 of oder n, 382 Lebesgue integral, 367, 368 of oder n (non-homogeneous), 383 Lebesgue integral properties, 368 of oder n with constant coefficients, 383 Lebesgue lemma, 469 with periodic coefficients, 716 Lebesgue measure, 365, 366, 486 Linear differential operator, 668 Lebesgue-negligible set, 368 Linear extension, 103 Ledrappier theorem, 731 Linear extension of a partial order, 45 Leech lattice, 9 Linear form, 105, 238 Lefschetz fixed point theorem, 304 Linear function, 20 Lefschetz number, 304, 571 Linear functional, 20 Lefschetz theorem, 304 Linear group", 240 Left adjoint, 593, 607 Linear independence, 102 Left adjoint and right adjoint functor, 302 Linear interpolation method, 340 Left adjoint functor, 595 Linear Lie group, 131 Left and right distributivity, 43, 516 Linear map, 103 Left and right unitor, 586 composition, 105 Left cancellable element, 33 operations, 104 Left closed monoidal category, 587 Linear operator, 20, 372 Left derived functor, 557 Linear optimization, 670 Left exactness, 317 Linear order, 45 Left field, 79 Linear programming, xviii, 669 Left ideal, 22 Linear stability loss in dissipative systems, 718 Left projective module, 554 Linear topological space, 20 Left R-module M, 517 Linear topology, 533 Left resolution, 556 Linear transformation, 20, 158 Left surjective and univocal relation, 29, 519 Linear trend, 650 Left surjective relation, 29, 519 Linear variety, 513 Left translation, 82 Linearization, 685 Left zero morphism, 599 Linearization of (statistical) nonlinear models, 637 Left-exact functor, 595 Linearization of nonlinear model, 684, 685 Left-inverse, 32 Linearly dependent vectors, 181 Left-inverse element, 42, 515 Linearly independent vectors, 181 Left-invertible, 32 Linearly ordered set, 21, 44 Left-neutral element, 32, 42, 515 Lines in sheaf, 152 Left-shift, 707 Linked variable, 2 L eg, 178 Liouville boundedness theorem, 433

798 Index

Liouville theorem, 463, 465, 695, 716 Lower box-counting dimension, 730 Lipschitz, 229 Lower capacity, 730 Lipschitz condition Lower integral, 350 global, 386 Lower limit of sequence, 323 lo cal, 386 Lower pointwise dimension, 731 Little Desargues theorem, 155, 157 Lower set, 23 Littlewood theorem, 493 Lower triangular, 110 Ljung-Box statistic, 643 Lower triangular matrix, 109 LM tes, 630 Loxodrome, 180 LM test, 634 pseudospherical, 180 p Lobachevsky geometry, 152 L -function, 392 p Lobachevsky-Bolyai-Gauss geometry, 152 L -space, 392 p Local basis, 543 L [a, b] space, 371 Local bifurcation, 711 Lumped parameter model, 739 Local bifurcations, 734 Lune, 179 Local bifurcations inearby steady states, 734 Lusternik-Schnirelmann theorem, 294 Local bifurcations nearby a periodic orbit, 736 Lyapunov, 687 Local canonical parametrization, 432 Lyapunov characteristic number, 728 Local chart, 451, 708 Lyapunov dimension, 731 Local compactness and paracompactness, 540 Lyapunov exponent, 727 Local continuity, 50, 51, 200 Lyapunov function, 759 in terms of open balls, 51 Lyapunov stability, 718, 719 in terms of sequences, 51 Local coordinate chart, 541 m-fold cover, 307 Local coordinates, 413, 687 m-sheeted covering space, 307 Local cross section, 691 m-simplex, 278 Möbius function, 40 Local diffeomorphism, 454, 546 Möbius inversion formula, 40, 116 Local element (local section), 581, 610 Möbius strip, 412 Local embedding, 545 Möbius transformation, 439, 483 Local extrema, 331, 332, 345 MA (moving average), 645 Local extrema of functions from Rn to R, 345 MA model, 645 Local extremum, 346 M A (q), 645 Local family, 711 Mac Lane, 575, 589 Local homeomorphism, 306, 413, 538 Mac Lane coherence theorem, 586, 587 Local maximum, 45, 679 Mac Lane introduction to categories, 589 Local minimum, 679 Mackey topology, 376 Local presentation, 553 Madaurin series, 90 Local representatives, 708 Magma, 41, 515, 520, 521 Local ring, 526 Main convergence criterion, 323 Local section, 563, 581, 610 Mainardi-Codazzi relations, 405 Local stability, 758 Major axis, 185 Local stable manifold, 720 Majorant, 45, 46, 53, 370, 490 Local structural stability, 703 Majorization, 324 Local surface, 415, 542 Malgrange theorem, 711 Local system, 312 Malgrange-Ehrenpreis theorem, 669 Local system of groups, 312 Manifold, 300, 413—415, 513, 540, 708 Local trivialization, 561 Manifold with boundary, 417, 541 Local-global bifurcation, 747 Mantissa, 624 Local-ringed space, 270 Many-sorted language, 512 Locality axiom, 551, 606 Many-sorted logic, 512 Localization, 528 Many-to-one relation, 14, 29 Localization of a ring, 528 Map, 9, 29, 38, 519 Localization of an R-module, 528 Bijective, 31 Locally 1-connected space, 308 bijective, 29, 35, 519 Locally arcwise-connected space, 204, 546 contraction, 229 Locally compact Hausdorff space, 299 fixed-point, 229 Locally compact space, 210, 224, 422 injective, 29, 31, 519 Locally connected space, 203, 227 of the first return, 722 Locally constant sheaf, 552 surjective, 29, 31, 36, 519 Locally exact form, 448 Map fixed point, 229 Locally finite family of parts, 211, 496 Map germ, 552 Locally finitely-presentable flat morphism, 569 Map locally of finite presentation, 610 Locally invertible function, 345 Map object, 580 Locally Noetherian scheme, 564 Map sink, 724 Locally of finite type, 566, 569 Map winding number, 742 Locally path connected, 277 Mapping cone, 303 Locally pathwise-connecte, 539 Mapping cylinder, 277 Locally pathwise-connected, 538 Mapping fiber, 313, 314 Locally pathwise-connected space, 196 Mapping space, 233 Locally quasi-finite morphism, 569 M aps, 9, 29 Locally ringed space, 270, 552 Maps and structures for sets, 27 Locally small category, 582, 583, 585 Maps compatible with structures (morphisms), 38 Locally stable equilibrium, 757 Maps on unit circle, 740 Locally trivial bundle, 268, 315, 561 Marginal optimality conditions, 679 Location vector, 182 Markov chain, 616, 623 Locking, 743 Markov process, 616, 623 Locus, 71, 397, 531 Mass point, 190 Logarithm, 336, 455 Matching, 152 Logarithm in C, 76 Mathematical expectation, 621 Logarithmic barrier function, 674 Mathematical induction, 49 Logarithmic potential, 490 Mathematical logic, xvii, 1 Logic, xvii, 1, 512 Mathematical programming, 673 Logic of orders, 3 Mather theorem, 705 logic of predicates by level, 3 Matrix, 103, 104, 133 Logical connective, 1 Matrix diagonalization, 134 Logical constant, 3 Matrix form, 628 Logical propositions, 1 Matrix minimal polynomial, 134 logistic curve, 638 Matrix of affine collineation, 172 Logistic map, 715, 737 Maximal atlas, 542 Logistic model, 638 Maximal differentiable atlas, 540 Logistic model (Vehulst or Pearl curve), 638 Maximal element, 22, 35, 45, 46, 579 Long exact sequence, 289, 554, 605, 606 Maximal filter, 19 Long exact sequence of a pair axiom, 597 Maximal ideal, 22, 269, 528, 569 Long exact sequence theorem, 317 Maximal ideal (for lattices), 23 Long memory phenomena, 652 Maximal spectrum, 528 Long memory process, 650 Maximal tree, 297 defined by autocorrelation, 650 Maximum element), 45 defined by spectral density, 651 Maximum likelihood, 635 Long range dependence (LRD), 652 Maximum on a compact interval, 680 Longitudinal data, 633 Maximum principle, 466, 482, 486, 491 Loop, 214, 430, 444, 574 Maxwell-Faraday equation, 364 Loop space, 311 Mayer-Viétoris, 502 Lorentz curve, 663 Mayer-Victoris corollary, 301 Lorentz tensor, 409 Mayer-Vietoris corollary, 296 Lorentz transformation, 409 Mayer-Vietoris sequence, 296 Lorenz model, 749 Mayer-Vietoris theorem, 292 Lorenz system, 715, 716, 721 Mayer-Vietoris theorem for reduced homology, 292 Lotto, 615 M eagre, 232 Lower (Riemann) integral, 350 Meagre part, 232 Lower bound, 46 Meagre set, 232

799 Index

Mean curvature, 403, 419, 421 Models with errors on variables, 637 Mean curvature of a surface S2, 421 Modified Jordan form, 758 Mean lag, 640 Module, 38, 43, 81, 101, 103, 517 Mean square contingency, 623 calculation rules, 101 Mean value inequality theorem (for polynomials), 115 Module homomorphism, 101, 103 Mean value theorem, 331 Module of Kähler differentials, 527 Mean value theorems, 331 Module theory, 81, 101, 514 Mean-value formula, 456 M o dulo, 84 Mean-value property, 466 M o dulus, 71 Mean-value theorem, 332 Modulus of a complex number, 71, 76 Measurability, 193 Modus barbara, 2 Measurable function, 19, 367, 368 Modus ponens, 2, 5 Measurable set, 18 Modus tollens, 2, 5 Measurable space, 18 Moivre formula, 75 Measure, 18, 349, 442 M onad, 593 Measure concentrated on the attractor, 724 Monad (triple), 593 Measure determination, 350 Monadic, 511 Measure of hypographs, 349 Monadic adjunction, 593 Measure of length, 448 Monge, 150, 173 Measure of regular surface, 359 Monge-Ampère equation, 389, 391 Measure of surface, 359 Monic divisor, 110, 147 approximation by lateral surfaces of prims, 359 Monic generator, 111 approximation by polyhedrons, 359 Monic irreducible polynomial, 121 Measure of unbounded parts, 367 Monic polynomial, 36, 109, 120, 134, 563, 569 Measure theory, xvii, 365, 368, 442 Monic polynomial (unit polynomial, unitary polynomial), 36 Measurement, 160 Monkey saddle, 402 M edian, 166 Monodromy, 427 Mellin transformation, 657 Monodromy group, 428 Melnikov function, 710 Monodromy group of regular covering space, 308 Melnikov method, 710, 739 Monodromy matrix, 723 Mendelson, 1 Monodromy theorem, 306, 427 M enger, 213 Monogenic group, 83 M enger n-arc theorem, 262 Monoid, 42, 515, 517, 520, 522, 582, 593 Menger theorem, 262 Monoid (in category theory), 593 Mergelyan theorem, 479 Monoid (semigroup with identity), 590 Meromorphic continuation, 465 Monoid axioms, 520, 576, 582 Meromorphic function, 430, 465 Monoid homomorphism, 521 Meromorphic function on C, 434 Monoidal category, 586, 587 Meromorphy, 421 Monoidal identity object, 588 Meromorphy of C-C-functions, 422 Monoidal product, 586 É É Mersenne number, 148 Monoidal structure, 588 Meshing, 171 Monomial, 518 Metacategory, 590 Monomial algebra , 518 Metagraph, 590, 600 Monomorphism, 38, 264, 520, 598 Metamathematics, 4 Monomorphism (monic), 598 Metastable chaos, 751 Monomorphism (or monic), 606 Method of continuity, 390 Monotone arithmetic function, 141 Method of penalty functions, 673 Monotony criteria, 323 Method of proxy variables, 637 Monotony criterion, 141 Method of successive approximations, 390 Monster peaks, 650 Method of superposition, 624 Monte-Carlo method, 624 Metric, 20, 199 Montel theorem, 458 Euclidean, 199 M o ore, 2 p-adic, 79 Moore-Smith sequence, 206 Metric dimensions, 730 Morera, 426, 461 Metric entropy, 727 Morera theorem, 461 Metric geometry on a regular patch, 401 Morgan law, 9 Metric plane, 151 Morgan laws, 2, 6, 12 axioms M1,M2,M3, 151 Morphism, 33, 38, 576, 598 Metric projective plane, 158 bijective, 38, 598 Metric space, 50, 193, 199, 216, 220, 227 surjective, 38, 598 countable product, 217 Morphism locally of finite presentation, 566 finite product, 217 Morphism locally of finite type, 566 precompact, 229 Morphism of finite presentation, 566 totally compact, 229 Morphism of finite type, 566 uncountable product, 217 Morphism of schemes, 270, 563 Metric space dimension, 213 Morphism with finite fibers, 566, 567 Metric subspace, 217 Morse surgery, 373 Metric tensor, 406—408, 418 Morse theory, 373 Metric tensor (Riemannian metric), 407 Morse-Kelley set theory (MK), 15 Metric topology, 199 Morse-Smale systems, 703, 724, 748 Metrizable space, 210 Mother wavelet, 665, 666 Metrization, 193, 210 Multi-scale analysis, 666 Metrization theorems, 210, 211 Multicellular domain, 441 Meusnier theorem, 402 Multicollinearity, 631 Midpoint, 158 Multicollinearity problems, 632 Milne process, 388 Multifunctor, 595 M ilnor, 715 Multilinear form, 407 Minimal element, 10, 45 Multilinear map, 239 Minimal polynomial, 117, 122 Multiple, 141 Minimal surface, 373 Multiple correlation coefficient, 627, 629 Minkowski distance, 220 Multiple integral, 355 Minkowski inequality, 220, 242 Multiple regression model, 628 Minkowski metric, 409 Distinction between εt and et, 628 Minkowski norm, 178 estimation of estimators, 628 Minkowski tensor, 409 estimation of regression coefficients, 628 M inor, 106 general linear model, 628 Minor axis, 185 matrix form, 628 Minor Hessian, 346 geometric interpretation, 631 Minorant, 46, 370 properties, 628 Minorization, 324 properties of estimators, 629 M inus, 8 statistical tests, 629 Misspecification of model, 633 stochastic assumptions, 628 Mittag-Leffer theorem, 481 structural assumptions, 628 Mittag-Leffler decomposition, 434 variance analysis, 629, 630 Mixed product, 183 Multiple root, 112 Mixed structure, 37 Multiple roots, 121 Mixed tensor, 408 Multiple structure, 37, 38 Mixing dynamical system, 725 Multiple-valued complex function, 78 Moëbius meshing, 171 Multiple-valued function, 78, 431 Moëbius strip, 255, 400 Multiplication in N, 52 locally orientable, 400 Multiplication in Q, 61 Mode locking, 742 Multiplication in Z, 59 M o del, 303 Multiplicative associativity, 43, 516 Model (in logic), 510 Multiplicative identity, 42, 43, 516 Model (Relation system), 510 Multiplicative inverse, 42 Model theory, 509, 510 Multiplicative notation, 32, 39 Model validation, 625 Multiplicative set, 528 Model without constant term, 626 Multiplicative subset, 528 Models of product life cycle, 638 Multiplicatively stable, 40, 93

800 Index

Multiplicity, 462 Nevanlina class, 463 Multiplicity of a covering space, 307 New Foundations (NFU,NF), 16 Multiplicity of a root, 112 N ew ton, 14 Multiplicity of a zero, 462 Newton sums, 114 Multiplicity order, 134 Newton’s second law, 360 Multiplier, 748 Newton-Gregory method, 339 Multipliers, 681 Newton-Raphson method, 340 Multiview orthographic projections, 174 N ew ton m eter, 360 Newtonian· attraction, 347 n-ary operation, 510 N F , 16 n-ary relation, 28 N F U , 16 n-ball, 271 Nilpotent, 93, 135 n-ball, 215 Nilpotent element, 529 n-boundary, 286, 287 Nilpotent endomorphism, 135 n-cell, 299, 534, 585 n Nilpotent sheaf of ideals, 570 n-cell e , 299 Nilpotent sheaf of ideals„ 569 n-chain, 286 Nilpotent upper triangular matrix, 135 n-chart, 413 Nilradical, 529 n-coboundary, 317 Nine-point circle, 166, 167 n-cochain, 317 No middle third set, 730 n-cocycle, 317 No-retraction theorem, 472 n-cycle, 86, 286, 287 k Nodal point, 381 n-dimensional C -atlas, 541 Node, 258, 381 n-dimensional Grothendieck cohomology groups, 550 attracting, 717 n-dimensional holes, 550 center, 717 n-dimensionalhomotopy groups, 249 sink, 717 n-dimensional manifold, 540 Node (vertex), 258 n-dimensional Riemannian geometry, 415 Node saddle bifurcation, 718 n-disk, 271 Noetherian, 35 n-disk, 215 Noetherian affine scheme, 564 n-disk (or n-ball), 271 Noetherian induction principle, 35 n-F system (system with n degrees of freedom), 694 Noetherian ring, 142, 523 n-gon, 299 Noetherian scheme, 563 n-manifold, 300, 540 Non denumerability, 34, 35 n-order predicate calculus, 3 Non denumerability of R, 35 n-simple space, 313 Non Euclidean metric, 153 n-simplex, 286 Non parametric model, 685 n-skeleton, 535 Non-algebraic functions, 336 n-sphere, 271 Non-Archimedean valuation, 146 n-th homology group, 550 Non-autonomous differential equation, 716 n-times continuously differentiable function, 330 Non-bicontinuous, 194 n-times differentiable function, 330 Non-connected space, 202 n-tuple, 12 Non-constructibility theorem, 130 n-variables function, 30 Non-contradiction Naïve style, 581 in an axiomatic system, 5 Nabla, 343, 347, 348 Non-contradiction problem, 4 Nagata-Smirnov metrization theorem, 212 Non-degeneracy conditions, 699 Naive set theory, 8, 13 Non-Euclidean geometry, 155 Napier analogies, 180 Non-Euclidean orthogonality, 158 Napier number e, 323 Non-Euclidean plane, 154 Napier rules, 180 Non-hyperbolic fixed points for planar fields, 702 N app e, 188 Non-hyperbolic singular point, 699 Natural equivalence, 302 Non-hyperbolic singularity, 699, 701 Natural exponential, 336 of saddle-node type, 701 Natural measure, 725 Non-incident center, 158 Natural mumber, 62 Non-isolated singularity, 433 operations, 62 Natural number, 33, 62 Non-isotropic line, 158 Non-orientable surface, 255 addition, 63 associativity, 63 Non-principal filter, 19 commutativity, 63 Non-strict ordering, 45 Distributivity, 63 Non-structured set, 47, 49 monotony, 63 Non-topological map, 194 neutral element, 63 Non-trivial ideal (for lattices), 23 Simplification rule, 63 Non-wandering point, 690, 724 multiplication, 63 Noncommutative domain, 518 associativity, 63 Noncommutative ring, 523 commutativity, 63 Nondegenerate point, 315 Distributivity, 63 Nonhomogeneous integral equation of the second kind, 374 monotony, 63 Nonhomogeneous system, 107 neutral element, 63 Nonlinear diffeomorphism, 697 Natural number object, 581 Nonlinear dynamical system, 697 Natural perspective, 156 Nonlinear dynamical system , 717 Natural topology, 193, 197—199, 204, 413 Nonlinear estimation methods, 638 Natural transformation, 267, 302, 576, 600, 612 Nonlinear flow, 697 Naturally equivalent functors, 302 Nonlinear model, 637 Naturally equivalent functors (isomorphic functors), 302 Nonlinear optimization, 673 Naturally isomorphic , 597 Nonlinear programming, xviii, 669, 670, 673 Necessary and sufficient condition, 5 Nonlinearities, 742 Necessary condition, 5 Nonmeasurable set, 18 Necessary condition (NC), 5 Nonresidue, 144 N egation, 8 Nonsingular matrix, 105 Negation rules, 3 Nontrivial subharmonic function, 491 Negative autocorrelation, 633 Norm, 20, 146, 236, 708 Negatively correlated variables, 625 Norm of a complex number, 71 Negatively invariant set, 690 Norm of the convergence in quadratic mean, 244, 245 Negligible part, 232 Norm of the uniform convergence, 233, 237, 244, 245, 370, 371 Negligible set, 366 Norm of vector, 181 Neighborhood, 50, 193, 194, 198, 216 Norm on a vector space, 236 neighborhood, 199 Normability, 20 Neighborhood axioms, 197, 218 normable, 20 Neighborhood basis, 199, 200, 543 Normable topological vector space, 20 Neighborhood basis (local basis), 543 Normal component, 401 Neighborhood filter, 222, 543 Normal curvature, 401, 403, 419, 420 trace, 222 Normal curvature of a regular surface, 420 Neighborhood system, 193, 538, 543 Normal derivative, 489 Neighborhood system (neighborhood filter), 543 Normal distribution, 616, 617, 622 Neighborhood system basis, 211 Normal endomorphism, 137 Neil parabola, 395 Normal equation, 184 Neimark-Sacker theorem, 737 Normal equations, 626, 628 Nerve, 296, 535, 559 Normal factor, 89 Nerve of a cover of a simplicial complex, 296 Normal form, 128, 296, 734 Nested intervals, 323 Normal form theorem (differential equation), 698 Nested segments, 67 Normal forms for diffeomorphisms, 699 N et, 206 Normal forms of bifurcations, 733 Neutral element, 32, 40, 42, 43, 64, 82, 86, 515, 516 Normal monomorphism, 598 Neutral element for , 40 Normal morphism, 584, 598 Neutral line, 170 ⊤ Normal parameterization, 393 Neutral on the left, 32 Normal parameterization of a circular helix, 393 Neutral point, 170 Normal plane, 394 Neutrally stable, 687 Normal representation, 77

801 Index

of a complex number, 76 Open disk, 71, 194 Normal representation of complex numbers, 76 Open elementary chain, 260 Normal section, 402 Open Eulerian chain, 260 Normal series, 89 Open half-ball, 230 Normal space, 193, 206, 207, 222 Open image, 466 Normal subgroup, 84, 88, 297, 308, 598, 599 Open immersion, 567, 571 Normal to a patch, 399 Open interval, 194 Normal to a plane, 164 Open manifold, 542 Normal tower, 89 Open map, 201, 537 Normal vector, 397 Open mapping theorem, 241, 372, 472, 477, 482, 537 Normality hypothesis of errors, 627 Open morphism, 565 Normality of errors, 626 Open part, 216 Normality tests, 643 Open q-simplex, 294 Jarque-Bera test, 643 Open set, 50, 193, 195, 198, 199, 536, 674 Skewness and Kurtosis tests, 643 Open set axioms, 218 Normalizer, 308 Open set topology, 536 Normally independently distributed (n.i.d.), 643 Open subscheme, 564 Normally summable family, 240 Opening angle, 184 Normed algebra, 24 O p erad, 525 Normed linear space, 21 Operand, 512 Normed space, 20, 21, 370, 371 Operation table, 41, 515 Normed vector space, 21, 370 Operations in Gauss complex plane, 76 Normed vector spaces, 236 Operations in N, 52 North pole, 271 Operations in Z, 58 Not topologically pairwise arc-connected, 196 Operations on ordinals, 49 Notion of attractor, 724 Operations on sets, 30 Nowhere dense, 232 Operator, 20, 372, 375 ntersection of a simple arc with a plane, 396 compact, 375 nth (singular) homology group, 287 differentiable, 372 nth (singular) homology group of a space, 287 differential, 374, 377 nth cohomology group, 317 finite rank, 375 nth cyclotomic polynomial, 116 Fredholm, 376 nth homology group, 287, 304 Hermitian integral, 377 nth homology group of a chain complex, 290 integral, 374, 377 nth homology group of a complex, 290 invertible, 372 nth primitive roots of unity, 116 linear, 372, 416 nth relative cobomology group, 317 Operator algebra, 375 nth relative homology group Hn(X, A), 291 Operator set, 510 nth roots of unity, 131 Operator spectrum, 24 Nth unit roots, 656 Operator variable, 3 nth-rank tensor, 407 Opposite, 40, 64 in m-dimensional space, 407 of a complex number, 76 Null element, 9 of an irrational, 66 Null hypothesis H 0, 617 Opposite category, 265, 584 Null object, 580 Optimal mix of explanatory variables, 631, 632 Null sequence, 522, 534 Optimality condition, 669 Null vector, 181 Optimization, xviii, 669 Null-homotopic, 274 generalization, 671 Null-homotopic map, 274 normal form, 671 Nullary operation, 265, 510 Optimization theory, 669 Numéraire goods, 760 Optimization with two variables, 670 N umb er Optimum with constraint algebraic, 78 examples, 681 b-adic, 68 geometric interpretation, 681 Bernoulli, 337 Optimum without constraint, 680 B etti, 257 O rbit, 306 com plex, 69 Orbit space, 309 connectivity, 257 Orbital stability, 719 connectivity , 261 Order, 159, 579 edge, 263 Order and orientation, 159 Fermat, 131, 148 Order contact, 396 hypercomplex, 35 Order diagram, 45 irrational, 68 Order filter, 23 Mersenne, 148 Order ideal, 23 p-adic , 78, 79 Order isomorphic, 47 prim e , 147 Order of a group, 82 transcendental, 78, 122 Order of a polynomial , 110 Number of crosscaps, 300 Order of a root, 112 Number of edges, 296 Order of integration, 648 Number of elements of a set, 7 Order of integration of time series, 648 Number of faces, 296 Order of multiplicity, 134 Number of handles, 300 Order of ramification, 146 Number of vertices, 296 Order of the zero, 118 Number system Order relation, 7, 9, 44 axiomatic construction, 78 cyclic, 160 Number theory, xvii, 141 strict, 44 Numerable Cech cohomology, 559 Order relation on Q, 61 Numeration in base b, 56 Order relation on Z, 59 Numerical harmonic analysis, 655 Order structure, 38, 44, 63—65 Fast Fourier transformation (FFT), 656 antisymmetry, 44 formulas for trigonometric interpolation, 655, 656 reflexivity, 44 Numerical resolution of equation, 340 transitivity, 44 O b, 576 Order structure of R, 321 Object, 38, 310, 576, 579, 598 Order type, 47 Objective function, 670, 671, 681 comparison, 48 Observed residual, 626 Ordered directed set, 589 Obstruction, 315 Ordered field, 38 Odd polynomial, 109 Ordered set, 38, 44 OLS estimator, 633 lexicographic product, 49 ω-assignment, 2 Ordering relation, 7, 9 One-form, 409, 410 Ordinal, 47, 48 One-point compactification, 78, 210, 312, 422 fi nite, 48 of complex numbers, 78 infi nite, 48 One-relator group, 298 Ordinal classe, 49 One-sector neoclassic growth model, 757, 758 Ordinal comparison, 49 One-sided convolution, 662 Ordinal number, 33, 48 One-sided Fourier transformation, 657 Ordinary differential equations One-sided surface, 255 second-order, 717 One-sided test, 617 Ordinary geometry, 150 One-tailed test, 617 Ordinary least squares (OLS), 626, 644 One-to-one correspondence, 33 Ordinary metric projective plane, 158 One-to-one function, 29 Ordinary sequence, 322 One-to-one relation, 14, 29 Orientable differentiable manifold, 416 Open affine cover, 571 Orientable regular curve, 447 Open and closed maps, 537 Orientable surface, 255, 400 Open ball, 194, 197, 199, 216 Orientation, 159, 286, 363, 446 Open chain, 260 Orientation of a Jordan arc, 393 Open cover, 208, 551, 607 Orientation of a plane, 150, 399 Open covering, 193, 208, 499 Orientation of a surface, 399, 400 Open curve, 214 Orientation of ∆n, 286

802 Index

Orientation-preserving, 159, 162, 173, 187, 740 of the cotangent, 434 Orientation-preserving (reversing), 697 of the function gamma Γ, 434 Orientation-preserving diffeomorphism, 692 Partial graph, 258 Orientation-preserving homeomorphism, 688 Partially ordered group, 512 Orientation-reversing, 173 Partially ordered set, 9, 21, 45, 512, 579 Oriented angle, 75 Partially ordered set (Poset), 21, 45, 579 Oriented graph, 259 Partially ordered set (poset), 9, 45 Oriented plane, 159 Particular first-order differential equations, 379 Oriented plane as topological space, 159 Partition, 8, 9 Oriented regular path, 446 Partition of unity, 411, 481, 495, 499 Oriented simplicial complex, 295 Partitions and covers, 32 Orientibility, 399 Partitions of an interval, 349 Origin of a path, 279 Pascal relation, 55 Origin of edge of simplicial complex, 297 Pascal triangle, 333 Orthocenter, 167 Pascal’s arithmetical triangle, 55 Orthodrome, 179, 180 Patch, 415, 542, 687 Orthogonal, 242 Patch of class r , 398 Orthogonal affinity, 168 Patch point, 402C Orthogonal axonometric map, 174 elliptic point, 402 Orthogonal basis, 137 hyperbolic point, 402 Orthogonal collineation, 165 parabolic point, 402 Orthogonal coordinate system, 173 planar point, 402 Orthogonal curvilinear coordinates, 343 Path, 196, 204, 260, 276, 392, 444, 499, 574 Orthogonal group, 131, 187 regular, 444 Orthogonal group On, 172, 191 Path class, 279 Orthogonal measure, 479 Path component, 276 Orthogonal of a part of a pre-Hilbert space, 242 Path connected, 276 Orthogonal projection, 174, 243 Path connected space, 276 Orthogonal system, 191, 243, 666 Path lifting property, 574 Orthogonal vectors, 191 Path of integration, 660 Orthogonal wavelet, 666 Path-connected, 546 Orthogonality, 21, 158, 242 Path-connected (0-connected), 546 according to a hermitian form, 136 Path-connected set, 196 Orthogonality condition, 400 Pathwise connected set, 196 Orthographic projection, 174 Pathwise connectedness, 546 Orthonormal algebraic basis, 244 Pathwise-connected, 196, 538, 539, 545, 546 Orthonormal basis, 137 Pathwise-connected (path-connected, 0-connected), 546 Orthonormal coordinate system, 172, 182 Pathwise-connected space, 196 Orthonormal part of pre-Hilbert space, 137 Paving, 447 Orthonormal system, 191, 243 Peak of frequency, 650 Oscillatory, 686 Peano, 4 Osculating circle, 394, 396, 398 Peano axioms, 49 Osculating plane, 394 5th, 5, 49 Osculating sphere, 396 Peano existence theorem, 386 Ostrogradski formula, 363, 501 Pearl curve, 638 Ostrogradsky formula, 411 Peixoto theorem (I) (for vector field, flow), 704 Ostrogradsky-Stokes formula, 363 Peixoto theorem (II) (for vector field, flow), 704 Ostrogradsky-Stokes theorem, 363 Peixoto theorem (III) (for diffeomorphism), 705 Ostrowski, 145 Pentagon diagram, 586 Output variable, 685 Percentage distribution, 620 Outset, 707, 709 Perfect field, 125, 530 Outward perturbation, 747 Perfect group, 315 Over-harmonic, 493 Perfect Reinhardt domain, 441 Over-harmonic function, 493 Perfect set, 369 Overlapping of regular patches, 399 Period bands, 436 -module, 565 OX Period parallelograms, 436 Period three implies chaos theorem, 733 p-adic absolute value, 145 Period-doubling (flip bifurcation), 737 p-adic distance, 220 Period-doubling (flip) bifurcation, 748 p-adic metric, 79 Periodic , 714 p-adic number, 67 p-adic number, 78, 79 Periodic complex function, 435 p-adic topology, 79 Periodic function, 435, 436 p-adic valuation, 79 complex, 435 p-admissible open set, 306 doubly, 436 p-function, 434 sim ply, 436 p-linear antisymmetric form, 139 Periodic non-autonomous system, 693 p-value (statistics), 617 Periodic orbit, 710 Pair category, 584 Periodic point, 687 Pairs of harmonic points, 172 Periodic point of an Anosov diffeomorphism, 705 Pairwise disjoint equivalence classes, 28 Periodic signal, 726 Pairwise disjoint subsets, 28 Periodic solution of a system, 717 Pairwise relatively prime, 57 Periods of complex function, 435 Palis-Smale theorem, 724 Permutation, 613 Panel analysis, 633 Permutation cycle, 86, 90 Panel data, 625, 633 Permutation group, 86 Pappus-Pascal theorem, 155 Permutation with repetitions, 614 Parabola, 185, 186 Permutation without repetition, 614 Parabolic fixed point, 747 Perpendicular bisector, 151, 153, 161, 167 Parabolic geometry, 149 Perron integral, 370 Parabolic point, 402, 403, 419 Perspective, 156 Paracompact space, 208, 211, 538 Perspective collineation, 156, 170 Paradoxical decomposition, 8 Perturbation, 685 Paradoxical set, 8 inward , 747 Paradoxical set with respect to the group G, 8 outward , 747 Parallel affine subspace, 190 Pesin formula, 728 Parallel axonometric projection, 174 Petals, 298 Parallel lines, 154 Pfaffian differential form, 408 Parallel postulate, 149 Pfaffian form, 408 Parallel projection, 168 Pfaffian form space, 408 Parallelepiped, 191 Phase diagram, 715 Parametering, 445 Phase plane, 715 Parameters of the model, 626 Phase portrait, 715, 716 Parametric representation, 393, 397, 398 Phase portrait of the flow, 690 Parametrization Phase spectra, 662 of a curve, 445 Phase-locking, 742 Parametrized arc, 392 Phase-space, 715 Parseval equality, 653 Phi function, 57 Parseval equation, 655 Phillips and Perron tests, 644 Parseval formula, 244 Phillips curve, 638 Part, 11 Physical depreciation rate, 760 bounded, 209 Physical measure, 725 denumerable, 54 P icard, 428 fi nite, 54 Picard iteration, 387 stable by a map, 30 Picard-Lindelöf theorem, 714 Partial autocorrelation function, 643 Piecewise differentiable functions, 659 Partial correlation, 631 Piecewise linear map, 295 Partial derivative, 342, 343 Piecewise monotonic function, 351 Partial differential equation, 388 Piecewise regular boundary, 501 Partial fraction decomposition, 335, 434, 659 P iecew ise- k path, 444 of a rational function, 434 P iecew ise-C r geometric arc, 392

803 Index

Pigeonhole principle, 53 Poset, 9, 45 Ping-pong lemma, 518 Position vector, 182 Pitchfork bifurcation, 736 Positive autocorrelation, 633 Pitchfork bifurcation (normal form), 733 Positive hermitian form, 137, 242 P ivot, 670 and Minkowski inequality, 137 Pivotal column, 669 and Schwarz inequality, 137 Pivotal row, 670 Positive homogeneity, 20 Planar affinities, 168 Positive measure, 442 Planar curve, 394 Positive scalability, 20 Planar geometric projection, 173 Positively correlated variables, 625 Planar graph, 259, 261 Positively invariant set, 690 Plane, 150, 159 Posterior distribution, 618 Plane curve, 214, 395, 397 Posterior probability, 615, 618 Plane lamina, 412 Power, 33, 34 Plateau function, 495, 500 Power functions with rational exponents, 336 Platonic solids, 296 Power object, 580, 609 Plucker coordinates, 138 Power object (in topos), 609 Poincaré, 4, 138 Power of the continuum, 37 Poincaré Bendixson theorem, 720 Power set, 8, 13 Poincaré circle model, 176 Power spectrum, 726 Poincaré conjecture, 295 Power theorem, 167 Poincaré cut, 721 Power-law, 652 Poincaré formula, 413 Prandtl number, 726, 749 Poincaré half-plane model, 177 Pre-Hilbert space, 21, 136, 137, 242, 370, 371 Poincaré lemma, 448, 498 Pre-Hilbert space (inner product space), 136 Poincaré map, 416, 692, 693, 695, 721 Pre-image, 521 for autonomous differential equations, 721 Pre-inner product, 137 for non-autonomous differential equations, 721 Pre-limit, 266, 602 roles, 721 Preadditive category, 585 Poincaré model, 176 Precompact metric space, 229 Poincaré recurrence theorem (Poincaré return theorem), 718 Precomposition, 365 Poincaré section, 717, 721, 722 Predecessor, 49, 51, 322 Poincaré theorem, 412 Predicate, 1 Poincaré-Bendixson theorem, 690 Predicate (with 3 arguments), 2 Poincaré-Brouwer theorem, 412 Predicate calculus, 2 Poincaré-Cartan invariant, 695 Predicate calculus theorem, 2 Point, 150 Predicate calculus theorems, 3 accumulation, 195 Predicate symbols, 511 adherent, 195 Predicate variable, 3 boundary, 195 Predicates, 2 cluster, 195, 322, 323, 341, 423 Preimage, 29, 53, 194, 519—521, 560 exterior, 195 Preimage (inverse image), 520 frontier, 195 Prelimit, 602 heteroclinic, 703 Preorder, 23 homoclinic, 703 Preordered set, 579, 582 interior, 195, 537 Preordering, 579, 582 isolated, 195 Presentation of a group, 297 lim it, 195 Presentation of an abelian group, 285 of equilibrium, 714 Presheaf, 267, 606 stationary, 714 Presheaf (algebraic topology), 551 Point at infinity, 157, 422 Presheaf (category theory), 551 Point-set topology, 193 Presheaf of groups, 267, 551 Pointed deformation, 313 Presheaves, 267, 551 Pointed homotopy axiom, 312 Primary decomposition, 136 Pointed homotopy category, 281 Prime element, 142 Pointed map, 272 Prime field, 125, 126 Pointed pair, 314 Prime ideal, 22, 100, 142, 143, 269, 528 Pointed pair homotopy, 314 Prime number, 56, 120 Pointed pair map, 314 decomposition, 147 Pointed space, 538 Prime spectrum, 529 Pointed topological space, 538 Prime subfield, 125 Poisson bracket, 698 P rim eval, 2 Poisson distribution, 617, 621 Primitive, 352, 353 Poisson formula, 486 Primitive element, 123 Poisson integral, 487 Primitive function, 349, 352 Poisson kernel, 486, 488 Primitive integral, 349 Poisson stability, 718 Primitive of a closed form along a path, 469 Poisson theory, 426 Primitive of a differential form, 498 Polar, 156, 184 Primitive polynomial, 121 Polar blowing up, 701 Primitive polynomial irreducible, 121 Polar coordinates, 76, 687 Primitive root, 125 Polar form, 136, 137 Primitive root of unity, 73 Polar plane, 184 Principal analytic set, 560 Polar topology, 377 Principal argument, 74, 76 Polar triangle, 154, 180 Principal bundle, 268, 561 Pole, 118, 152, 156, 158, 184, 428, 429, 464, 465 Principal curvature, 403, 420 Polish space, 232 Principal determination, 455 Polyadic, 511 of argument, 455 Polydisk, 440 of the logarithm, 455 Polygon, 168, 289, 669 Principal determination of the argument, 74 Polygon area, 168 Principal direction, 188, 189 Polygonal chain, 254 Principal G-bundle, 561 Polygonal group, 254 Principal ideal, 22, 101, 528, 529, 560 Polygons equal by adjunction, 169 Principal ideal domain, 101, 528 Polygons equal by decomposition, 168 Principal ideal ring, 101, 528, 560 Polyhedra, 250, 253 Principal normal, 373, 393, 398 as topological subspaces, 250 Principal normal vector, 394 Polyhedral Formula, 412 Principal order filter, 23 Polyhedron, 250, 251, 255, 294, 296 Principal order ideal, 23 Polyhedron in (Rp, R p), p V 3, 251 Principal part, 430, 465 Polynom ial Principal ring, 95, 101, 142, 528 Bernstein, 338 Principal symbol, 389 Polynomial degree, 120 Principal value, 499 Polynomial equivalence, 69 Principia Mathematica, 16 Polynomial function, 112, 333 Principia mathematica, 14 Polynomial function of lags (Almon’s lags), 640 Principle of analytic continuation, 457, 486 Polynomial interpolation, 339 Principle of excluded middle, 4 Polynomial map, 112 Principle of induction, 5 Polynomial ring, 69, 119, 120 Principle of isolated zeros, 462 Polynomial roots, 77, 112 Principle of recurrence, 5, 49 Polynomials Principle of transfinite induction, 47 Chebyshev, 338 Prior distribution, 618 Legendre, 339, 340 Prior probability, 615, 618 Polytope, 168, 250, 251, 255, 534 Probability, xviii, 613, 615, 618 Polytope edges, 251, 255 Probability calculation, xviii, 615 Polytope vertices, 251, 255 Probability charges, 26 Population, 616, 622 Probability density, 620, 634 Population variance, 616 Probability density function (p.d.f), 615, 620 Porter-Hudak, 651 Probability function, 620 Portmanteau test, 643 Probability generating function (p.g.f), 616

804 Index

Probability mass function (p.m.f), 615, 620 Proper map, 544 Probability measure, 615, 618 Proper morphism, 565 Probability measures on attractors, 724 Proper open set, 309 Probability sampling, 615 Proper subgroup, 83 Probability space, 615 Proper subset, 7, 13 Probable error (probability deviation), 617 Properties of a category, 256 Problem of enumeration, 53 Properties of a functor, 257 Pro cess Properties of an index, 471 A RC H , 652 dependence with respect to a point, 471 A R FIM A , 651 invariance by homotopy, 471 A R M A , 652 Properties of estimators, 626 DS (Difference Stationary), 644 Properties of limit values, 323, 325 FIGARCH, 652 Properties of Tor, 304 GARCH Properties preserved by topology change, 218 integrated, 652 Property P, 65, 66 IG A RC H , 652 Propositional and predicate Calculus, 2 long memory, 650 Propositional calculus, 2 of first order, 644 Propositional calculus theorems, 1, 2 TS (Trend Stationary), 644 Propositional formula, 1 Process of first order, 644 Propositions, 1 Process with memory, 633, 643 Proxy variable, 637 Product, 38, 603 Pseudo-convex, 441 Product category, 583, 584 Pseudo-random number, 624 Product group, 82 Pseudo-random numbers, 616, 623 Product law, 40 Pseudo-Riemannian metric, 409 Product metric space, 217 Pseudo-tensor, 408 Product of a family of complete normed spaces, 237 Pseudoconvex, 441 Product of a family of normed vector spaces, 237 Pseudoconvex domain, 441 Product of edge paths, 297 Pseudometric, 220, 230 Product of objects, 310 Pseudometrics and distances, 234 Product set, 13 Pseudosphere, 177 Product space, 202, 237 Pseudospherical loxodrome, 180 Product structure, 38 Pseudotensor, 408 Product topology, 66, 202, 237 pth exterior power, 316 Product uniform structure, 237 Pullback, 266, 305, 314, 364, 498, 604, 605, 607 Products of compact spaces, 224 Pullback (differential), 365 Programming Pullback (fiber product), 604 convex, 673, 674 Pullback (pullback diagram, fiber product, fibre product, fib ered product, discrete, 673 fiber square, fibre square, cartesian square, pullback square), 604 linear, 673, 674 Pullback as fibre-product, 365 mathematical, 673 Punctured plane, 78 nonlinear, 674 Puppe sequence, 313 quadratic, 673 Puppe sequence theorem, 313 stochastic, 673 Pure imaginary number, 69 Pro j, 572 Pure rotation, 691, 692 Proj as scheme, 572 Pure set, 14, 16, 578 Proj as set, 572 Pure set (hereditary set), 14, 16, 578 Proj as topological space, 572 Pure sinusoidal function, 726 Projection, 310, 561 Purely categorical, 581 Projection morphism, 603 Purely random process, 633 Projection of fiber space, 562 Pushforward, 19 Projection theorem, 21, 26 Pushforward (differential), 365 Projection theorems, 242 Pushforward measure, 19 Projection to normal plane, 395 Pushout, 266, 305, 535, 605 Projection to osculating plane, 395 Pushout (fibered coproduct, fibered sum, cocartesian square, pushout square, Projection to rectifying plane, 395 amalgamed sum), 605 Projective algebraic set, 513, 532 Pythagoras theorem, 242 Projective bi-ratio, 176 Pythagorean theorem, 169, 175, 178 Projective ccorrelation, 158 Pythagorean triangle, 93 Projective collineation, 156, 158, 171, 173 Pythagorean triplets, 144 Projective collineation in P 2 (R), 173 2 Projective correlation, 156 χ -distribution, 617 2 Projective general linear group PGLn(q), 172, 191 χ -test, 617, 622 Projective geometry, 149, 156 q-cell, 297 Projective group, 149 q-coboundary, 558 Projective map, 170, 171 q-cochain, 559 Projective metric, 158 q-simplex, 558 Projective module, 554 q-skeleton, 295 Projective morphism, 572 q-th Cech cohomology, 559 Pro jective n-space, 270 qth relative simplicial homology group, 296 Projective n-space on a ring, 572 qth simplicial homology group, 295 Projective Nullstellensatz theorem, 533 Quadrangle, 158 Projective object, 555, 556 Quadratic form, 678 Projective perspective, 170 definite negative, 678 Projective plane, 155, 157, 412 definite positive, 678 Projective plane of incidence, 155 semi-definite negative, 678 Projective polarity, 152, 158 semi-definite positive, 678 Projective resolution, 556 Quadratic programming, 670 Projective scheme, 513, 564 Quadratic reciprocity law, 144 Projective transformation, 156 Quadratic reciprocity theorem, 144 Projective variety, 513, 532 Quadratic residue, 144 Proklos or Proclus, 13 Quadratic surface, 186 Pro of, 6 Quadratic trend, 650 auxiliary hypothesis, 6 Quadric, 186 by construction, 6 Quadrics, 188 by contradiction, 6 classes, 188 by contrapositive, 6 coincident plane, 188 by descent, 6 ellipsoid, 188 by disjunction of cases, 6 elliptic cone, 188 by exhaustion, 7 elliptic cylinder, 188 by mathematical induction, 6 elliptic paraboloid, 188 by recurrence, 6 general equation by transposition, 6 principal directions, 188 combinatorial, 7 hyperbolic cylinder, 188 constructive, 6 hyperbolic paraboloid, 188 direct, 6 hyperboloid of 1 sheet; 2 sheet, 188 dissection, 7 intersectiong planes, 188 elementary, 6 parabolic cylinder, 188 nonconstructive, 6 parallel planes, 188 two-column, 6 Quadrilateral, 154 visual, 7 Quantifier, 2 without words, 7 Quantifiers, 2 Proof by mathematical induction, 5 Quantization, 2 Proof of excision, 293 Quasi-coherent sheaf, 553 Proper class, 16, 577, 578, 582, 585, 595 Quasi-coherent sheaf of ideals, 571 Proper divisor, 120, 141 Quasi-compact morphism, 566 Proper face, 294 Quasi-compact part, 208 Proper ideal, 22, 528 Quasi-compact space, 208, 538 Proper ideal (for lattices), 23 Quasi-difference transformation, 635 Proper ideal point, 157 Quasi-differences, 635

805 Index

Quasi-finite morphism, 566 Rational power function, 336 Quasi-inverse functor, 596 Rational rotation, 691 Quasi-isomorphism, 534, 592 Rational rotation number Quasi-projective variety, 533 (frequency locking), 744 Quasi-random number, 624 Ratios of distances, 169 Quasi-separated morphism, 565 Ray, 152, 159 Quasi-tensor category, 587 Rayleigh number, 749 Quasiperiodic function, 718, 726 Rayleigh-Bénard instability, 726 Quasiperiodic signal, 726 Rayleigh-Bénard phenomenon, 749 Quasiperiodic solution, 722 Rayleigh-Benard instability (R-B), 741 Quasiperiodicity, 734 Real analysis, xvii, 193, 198, 321 Quasiprojective variety, 513 Real analytic function, 548 Quasitensor category, 586 Real balances, 761 Quaternary relation, 161 Real differentiability, 424, 549 Quaternion, 35, 79 Real function, 324 Quaternionic projective n-space, 299 algebraic operations and composition, 325 Quotient, 64, 109 rational operations and composition, 326 Quotient category, 273 Real line, 216, 225 Quotient complex, 290, 298 not compact, 223 Quotient field, 38, 99, 117 Real line bundle, 561 Quotient group, 38, 88, 505 Real number, 65, 66 Quotient map, 574 algebraic structure, 66 Quotient module, 38 Real part, 70, 76 Quotient object, 580 Real part of z, 70 Quotient of the euclidean division of polynomials, 109 Real projective n-space, 299 Quotient relation, 38 Reciprocal circular function, 337 Quotient ring, 38, 43, 100, 143, 516, 526 Reciprocal hyperbolic function, 337 Quotient ring (factor ring, residue class ring), 526 Reciprocation, 157 Quotient ring (factor ring, residue-class ring), 43 Rectangle axiom (R), 153 Quotient set, 28, 34 Rectangular impulse, 658 Quotient space, 201, 275, 574, 575 Rectangular tiling, 447 Quotient space (identification space), 574 Rectifiable curve, 358 Quotient structure, 38 Rectifiable path, 499 Quotient topology, 201, 255, 275, 574, 575 Rectifiable set, 358 Rectification, 130 2 2 R -R -function, 423 Rectifying plane, 394 R2-R-function, 423 Recurrence, 47, 51 R-algebra, 523 Recurrence equation, 639 R-algebraic structure homomorphism, 521 Recurrence hypothesis, 5 R-analytic function, 486 Recurrence relation, 39, 51 R-B instability, 750 Recurrence theorem, 51 R-B phenomenon, 750 Recursive, 5 R-B problem, 751 Recursive definition, 5 R-integrability, 349 Recursive model, 625, 642 R-integrability of monotonic function, 351 Recursive regression, 631 R-integrable function, 349—351 Recursive residual, 631 R-module, 517, 521 Reduced algebra, 525 R-module homomorphism, 536 Reduced closed edge path, 297 R-module M, 517 Reduced edge path, 297 α-representatives, 708 Reduced equations, 641 R/S analysis, 651 Reduced form, 128 R/S analysis (Hurst statistic), 651 Reduced homology, 292 Rössler model, 729 Reduced homology groups, 292 Rademacher functions (square wave), 668 Reduced ring, 525 R adial, 491 Reduced scheme, 513, 564 R adian, 182 Reduced simplicial homology group, 296 Radical, 129, 529 Reductio ad absurdum, 6 Radical ideal, 529 Reduction theorem, 297 Radical of a polynomial, 115 Refinement, 567 Radical of ideal, 529 Reflection, 151, 161, 168 Radical or power center theorem, 167 Reflection with respect to a plane, 164 R adius, 150 Reflection-rotation, 165 Radius of convergence, 328 Reflection-translation, 162, 164 Radius of convergence , 333 Reflexive normed vector space, 239 Radius of convergence of series, 426 Reflexivity, 7 Radius of curvature, 177, 394, 396 Region having no holes, 545 Radius vector, 182 Regression on the first differences, 639 Radon measure, 442, 446 Regula falsi, 340 Raising to a power, 77 R egular Ral money stock per unit of labor, 760 curve, 445 Ramification point, 433, 437 path, 444 Random, 616, 623 Regular arc, 393 Random error (or noise), 616 Regular attractor, 724 Random in design of experiments, 616 Regular boundary, 500 Random number generator, 616, 623 Regular cardinal, 15, 578 Random numbers, 616, 623 Regular category, 588 Random process, 642 Regular covering space, 307 Random sample, 615 Regular curve, 445 Random variable, 616, 620 Regular domain, 356 Random vector, 616 Regular element, 33, 40 Random walk, 616, 623 Regular epimorphism, 588, 599 Random walk model, 644 Regular geometric arc, 393 Random with a uniform distribution, 623 Regular local ring, 526 Randomize, 616 Regular map, 531 Range, 14, 29, 519 Regular matrix, 105 R ank, 546 Regular matrix (nonsingular matrix), 105 Rank of a differential map, 546 Regular monomorphism, 599 Rank of a pure set, 15, 578 Regular n-sided polygon, 84 Rank of an abelian group, 286 Regular patch, 398 Ratio of affinity, 168 angle measure, 400 Ratio of similarity, 165 area measure, 400 Ratio of similitude, 165 curvature, 401 Ratio: chord length/circle diameter, 174 isometric, 400 Rational domain, 43, 516 length measure, 400 Rational expectations, 761 metric geometry, 401 Rational fraction, 119 Regular patches, 392, 398, 399 Rational function, 119, 333, 334 overlapping, 399 Rational Number Regular path, 444 calculation rules, 64 Regular ring, 526 Rational number, 64, 65 Regular space, 193, 206, 207, 221 algebraic structure, 65 Regular subharmonic function, 491 defined by alignments, 65 Regular submanifold, 544 negative, 65 Regular submanifold property, 544 order structure, 65 Regular surface, 359 default, 65 explicit representation, 359 positive, 65 parametric representation, 359 topological structure, 65 Regular T1-space, 207 default, 66 Regular value, 133 Rational numbers, 60 Regularity, 40 Rational or irrational ratio, 722 Regularity of the addition, 52

806 Index

Regularity of the addition and multiplication in N, 52 obtained by compactification, 78 Regularity of the multiplication, 55 of logarithm function, 432 Regularity of the multplication, 52 of sphere with handle, 432 Reinhardt domain, 421, 441 of square root function, 432 Relaminarization, 746 topological structure, 432 Relation, 2, 297 Riemann tensor, 409, 415 left surjective, 29, 519 Riemann zeta function, 338 less than or equal to, 36 Riemann-Christoffel curvature tensor, 409 univocal, 29, 519 Riemann-Christoffel tensor, 409 Relation of homotopy, 573 Riemann-integrable function, 350, 368 Relation symbols, 511 Riemann-Stieltjes integral, 369 Relational signature, 511 Riemannian connection, 418 Relational system, 510 Riemannian geometry, 413, 415 Relations, xvii, 27, 29, 38, 681 Riemannian manifold, 407, 415, 418, 540 compatible, 38 Riemannian metric, 407, 408, 418 conditions of compatibility, 38 Riemannian space, 415 Relations and structures, xvii Riemannian structure, 415, 708 Relations between coefficients and roots (of a polynomial), 113 Rienhardt domain, 441 Relative cobomology group, 317 Riesz decomposition theorem, 492 Relative complement, 8, 12 Riesz representation theorem, 21, 375, 442 Relative frequency, 620, 624 Riesz theorem, 375 Relative homeomorphism, 299, 300 Right adjoint, 589 Relative homology group, 291 Right adjoint functor, 595 Relative homotopy, 279 Right angled triangle, 178 Relative homotopy group, 314 Right cancellable element, 33 Relative Hurewicz theorem, 315 Right closed monoidal category, 587 Relative slmplicial homology group, 296 Right derived functor, 556, 557 Relative Spec, 572 Right ideal, 22 Relative tensor, 408 Right R-module M, 517 Relative topology, 197, 538 Right resolution, 556 Relatively compact, 480 Right resolution (coresolution, resolution), 558 Relatively compact set, 375, 458, 480 Right triangle, 175, 178 Relatively compact set (conditionally compact set), 375 Right zero morphism, 599 Relatively prime, 57, 143, 144 Right-exact functor, 595 Relatively prime (coprime), 111 Right-inverse, 32 Remainder, 109 Right-inverse element, 42, 515 Remainder of the euclidean division of polynomials, 109 Right-invertible, 32 Remarkable identity, 110 Right-neutral element, 32, 42, 515 Removable singularity, 464 Rigid motion, 161, 163 Renaming, 37 Rigid object, 588 Repeating decimal, 68 Rigid transformation, 161, 163 Representable functor, 267, 597, 600 Rigid transformation (rigid motion), 161 Representation, 580 Rigid transformation in R3, 164 Representation (representing object), 580 Ring, 38, 42, 81, 92, 516 2 Representation of a curve in R , 397 of integers, 63 explicit representation, 397 of polynomials, 69 implicit representation, 397 without divisor of zero, 43, 516 parametric representation, 397 R ing , 522 Representation of a diffeomorphism, 691 Ring homomorphism, 94, 99, 521 Representation of line, 183 Ring homomorphism (ring morphism), 94 equation with point and normal, 184 Ring isomorphism, 94 equation with vector product, 184 Ring morphism, 94 homographic equation, 184 Ring of congruence classes, 100 vector equation with point and direction, 183 Ring of fractions, 528 vector equation with two points, 183 Ring of integers, 63, 64 Representation theorem, 243 calculation rules, 64 Representation theory, 509 Ring of integers modulo n, 96 Representative, 28 Ring of matrices (n,n), 43, 516 Representing object, 580 Ring of polynomials, 43, 516 Reproducing kernel, 461 Ring spectrum, 529 Residual, 23, 626 Ring theory, 81, 514 R esidual et, 626 Ring unit, 526 Residue, 119, 144, 430, 476, 506 Ring with a one, 522 Residue class, 143 Ring with identity, 43, 516, 522 Residue class ring, 526 Ring with unity, 43, 516, 522 Residue classes, 100 Ringed space, 513, 552 Residue field, 270, 526 Ritt algebra, 527 Residue theorem, 430, 476 Rolle theorem, 331, 332 Residue-class ring, 43, 143 R o of, 601 Resolution, 555, 556, 558 Root extraction, 128 Resolution (abstract algebra), 558 Root nth of a complex number, 77 Resolution (homological algebra), 556 Root of unity, 73, 131 Resoluton, 553 Root of unity (de Moivre number), 73 Resoluton of differential sheaf, 553 Roots of a polynomial, 112, 121 Resonance, 698 Roots of a polynomial of degree n, 121 Resonant condition, 700 Roots or zeros of a polynomial, 112 Resonant eigenvalues, 698 R ose, 298 Restoring force, 648 Rose (bouquet of circles), 298 Restricted homotopy, 573 Rotation, 41, 157, 162, 164, 515 Restriction, 30, 573 of a regular polygon, 41, 515 Restriction of a map, 29, 573 Rotation group, 41, 131, 160, 191, 515 Restriction rule, 329 Rotation number, 691, 734, 740, 743 Retract, 249, 271, 573 Rotation of a vector field, 413 Retraction, 36, 271, 573 Rotation of angle, 173 Return force, 648 Rotational, 347 Return to scale, 758 Rotations, 157 Revealed preferences, 760 Rouché theorem, 477, 504 Reversal of axis, 173 Rough set, 16 Reverse homothety, 165 Routh-Hurwitz conditions, 753 Reynolds number, 727 Row stochastic matrix, 623 Ricci calculus, 407 RSA (cryptosystem), 96 Ricci curvature, 419 R uelle, 733 Ricci tensor, 409, 419 Ruelle-Takens theory (R-T), 727, 741 Richardson theorem, 3 Ruelle-Takens-Newhouse scenario (R-T-N), 727, 739, 741 R iem ann, 4 Rule of contraposition, 2 Riemann criterion, 350 Rule of modus barbara, 2 Riemann curvature, 415 Rule of modus ponens, 2 Riemann curvature tensor, 409 Rule of modus tollens, 2 Riemann geometry, 415 Ruled surface, 404 Riemann integral, 349, 350, 355, 368 Rules of distributivity, 2 of functions of several variables, 355 Rules of inference, 1 on measurable domains of R2 in the Jordansense, 356 R unge, 479 Riemann integral transformation into curvilinear integral, 363 Runge approximation theorem, 479 Riemann mapping theorem, 483 Runge theorem, 478 Riemann space, 407, 415 Runge-Kutta method, 388 Riemann sphere, 78, 415, 422, 423, 482 R ussel, 13 Riemann sum, 350, 358, 360 Russell, 8, 14 Riemann surface, 78, 415, 421, 423, 431 Russell paradox, 4 abstract, 432 Russell paradoxical set, 8 concrete, 432 Russell’s antinomy, 8

807 Index

Russell’s paradox, 8, 13, 14 Semiadditive category, 585 RVAR (Retricted Vector AutoRegressive), 647 Semigroup, 38, 42, 62, 515, 516, 520, 522 of natural mumbers, 62 σ-algebra, 18, 618 Semigroup homomorphism, 521 σ-compact space, 212 Semilocally 1-connected space, 308 σ-field, 18, 618 Semilocally simply connected, 308 Saddle, 717 Semilocally simply connected space (semilocally 1-connected space), 308 Saddle connection, 712, 746 Semimajor axis, 185 heteroclinic, 747 Semimetric, 230 homoclinic, 747 Semiminor axis, 185 Saddle connection bifurcation, 712 Sensitive dependence on initial conditions (SDIC), 730, 732 Saddle point, 381, 673, 694, 717 Sentential calculus, 2 Saddle-connection, 710 Sentential formula, 1 Saddle-connection bifurcation, 746 Sentential variable, 1 Saddle-node, 701, 748 Separable, 24, 211, 542 Saddle-node (fold) bifurcation, 748 Separable algebra, 530 Saddle-node and transcritical bifurcations, 734 Separable closure, 530 Saddle-node bifurcation, 711, 748 Separable element, 530 Saddle-node bifurcation (normal form), 733 Separable extension, 530 Saddle-node periodic orbit, 749 Separable morphism, 599 Saddle-node to a periodic-orbit, 738 Separable polynomial, 113, 125, 530 Saddle-point, 710, 760 Separable space, 24 Saddle-type diffeomorphism, 696 Separated morphism, 565 Same cardinality, 33 Separated parts, 202 Same homotopy type, 274 Separated presheaf, 608 Same orientation, 286 Separated scheme, 564 Sample, 615, 622 Separated space, 538 Sample space, 615 Separated space (Hausdorff space, T2-space), 538 Sample variance, 622 Separately holomorphic function, 494 Sampling distribution, 615 Separates points, 20 Sampling without replacement, 615 Separating hyperplane theorem, 26 Sampling witth replacement, 615 Separation axiom, 193 Samuelson oscillator, 751—753 T0 (Kolmogorov separation axiom), 207 SARIMA process, 645 T 1, 207 Satisfiable, 2 T2 (Hausdorff separation axiom), 207 Saturated set, 575 T3 (regularity axiom), 207 Savings rate, 760 T4 (normality axiom), 207 SBR measure, 725 Separation axioms, 205—207 SC criterion, 632, 647 T0,T1,T2,T3,T4, 206, 207 SC (p), 647 Separation of convex sets, 676 Scalar, 20 Separation of points, 474 Scalar curvature, 409, 419, 420 Separatrices, 747 Scalar curvature of a Riemannian manifold, 420 Separatrix, 747 Scalar density, 408 Separatrix loops, 721 Scalar multiplication, 181 Separatrix surface, 720, 723 Scalar product, 21, 182, 183, 244 Sequence, 30, 322 Scalar product on the space l2(K), 244 convergent, 67 Scalar square, 137 equivalent to zero, 66 Scale parameter, 666 in a metric space, 221 Scaling factor, 666 interleave, 216 Schauder, 370 monotone increasing , 66 Schematization of homotopy groups, 250 of Cauchy, 66, 67 Schem e, 563 Sequence convergence, 204 Schemes, 269 Sequence of complex numbers, 751 Schmidt orthogonalization process, 243—245 Sequence of holomorphic functions, 458, 477 Schoenflies theorem, 215 Sequential evolution law, 752 Schröder-Bernstein theorem, 34 Sequential space, 212 Schwartz, 391 Sequentially complete space, 231 Schwartz Laurent, 668 Series, 323 Schwartz theorem, 467 Series expansion, 332, 426 Schwarz criterion, 632 Series in normed vector spaces, 240 Schwarz criterion (SC), 640 Series of functions, 328 Schwarz function, 632, 640 Series with general term, 240 Schwarz inequality, 191, 242 Serration, 559 Schwarz lemma, 467 Serre fiber space, 314 Schwarz reflection principle, 461 Serre fiber space (weak fibration), 314 Scipion del Ferro formula, 74 Screw , 165 Serre theorem, 314 Seasonal adjustment, 645 Sesquilinear form, 136, 242 Secant, 163, 175 Set, 7 Secant of a circle, 163 at most denumerable, 35 Second axiom of countability, 538, 543 closed, 195 Second category, 232, 724 connected, 196 Second countable topological space, 199 denumerable, 35 Second countable topology, 199 finite or infinite, 34, 53 Second de Rham cohomology vector space, 502 frontier, 195 Second fundamental form, 401 non-denumerable, 35 patch theory, 401 open, 50, 195 Second order differential equation, 381 totally ordered, 44 Second sample central moment, 622 Set algebra, 8 Second-axiom of countability, 212 Set closure, 196 Second-countable, 211, 538, 539, 541, 542 Set difference, 8, 11, 13 Second-countable basis, 211 Set differences, 520 Second-countable space, 212, 543 Set direct product, 13 Second-order predicate calculus, 3 Set inclusion, 13 Section, 563 Set membership, 10 Section of a fiber bundle, 563 Set of accumulation point, 195 Section of Poincaré, 722 Set of adherent points (closure), 195 Segm ent, 67 Set of all hereditarily finite sets, 578 Segment division, 166 Set of all homogeneous prime ideals, 270 Seifert, 412 Set of all homotopy classes of pointed pair maps, 314 Seifert conjecture, 412 Set of all parts of, 7 Seifert-van Kampen theorem, 297, 298 Set of all rotations Selection, 19 of a regular n-sided polygon, 42, 516 Selector, 19 Set of all sets, 13, 14 Self-adjoint operator, 243 Set of all the sets, 8 Self-conjugate subgroup, 599 Set of antecedents, 53, 560 Self-homologous point, 166 Set of anti-hermitian matrices, 132 Self-similarity, 652, 731 Set of antisymmetric matrices, 132 Semantic antinomy, 13 Set of axioms, 5 Semantic method, 2 Set of definition, 119, 422 + Semantic or syntactic method, 2 Set of direct isometries: E (2), 163 Semi-continuous, 349 Set of distributions, 496 Semi-continuous function Set of ε-neighborhoods, 200 on Baire space, 233 Set of frontier points, 195 Semi-cubic parabola, 395 Set of generators, 102 Semi-distance, 220, 230 Set of individuals, 3 Semi-metric space, 227, 230 Set of inference rules (deductive reasoning), 5 separated, 230 Set of interior points, 195 Semi-norm, 20, 236 Set of invertible elements of a ring, 93 Semi-simple algebra, 494, 530 Set of majorants, 46

808 Index

Set of null-trace matrices, 132 Simple root, 112, 121 Set of parts of a set, 7, 31 Simple root of a polynomial, 121 Set of prime ideals, 269 Simplex, 190, 251, 255, 277, 534 Set of sesquilinear forms, 136 Simplex algorithm, 669 Set of the parts of, 14 Simplex method, 669, 671 Set of theorems, 5 Simplex of vertices, 252 Set of topologically equivalent points, 193 Simplex tableau, 669 Set of topologically non-equivalent points, 194 Simplexes, 251, 255, 277 Set theory, xvii, 7, 12 Simplexes in (Rp, R p), p V 3, 251 Set topology, 193 Simplices, 251, 255, 277 Set with one element, 579 Simplicial approximation, 253, 295 Set with regular boundary, 450 Simplicial approximation theorem, 295 Set-theoretic difference, 8, 12 Simplicial cohomology group, 319 Set-theoretic topology, 193 Simplicial complex, 250, 251, 255, 294, 534 Sets and algebraic structures, 81, 514 Simplicial complex in (Rn, R n), 253 Sharkovskii, 733 Simplicial homology, 251, 255, 257, 263, 295, 550 Sharkovskii ordering, 733 Simplicial homology functor, 597 Sharkovskii theorem, 733 Simplicial homology group, 295 Sharkovskii theorem (1964), 733 Simplicial k-chain, 550 Sharkovsky, 733 Simplicial map, 253, 295 Sharkovsky theorem, 733 Simplicial Mapping, 251 Sheaf, 153, 267, 551, 606, 607 Simplicial mapping, 263, 550 Sheaf (general topology), 551 Simplicial subcomplex, 535 Sheaf associated to a presheaf, 268 Simplicial subcomplexes, 251, 255 Sheaf axioms, 551, 606 Simplification rule, 63 Sheaf cohomology, 558, 570 Simply connected, 196, 422, 545 Sheaf cohomology by Cech cohomology, 558 Simply connected (1-connected), 545 Sheaf cohomology by derived functor, 558 Simply connected domain, 425 Sheaf of germs, 513 Simply connected open set, 470 Sheaf of germs of holomorphic functions, 560 Simply connected region, 197, 202, 425, 545 Sheaf of lines, 152, 153, 157 Simply connected region (1-connected region), 545 Sheaf of planes, 152 Simply connected space, 197, 202, 249, 282, 545 Sheaf of rings, 551, 552 Simply ordered set, 21, 44 Sheaf on a site, 608 Simply periodic function, 436 Sheafification, 571, 607 Simpson (approximation) method, 355 Sheafification (sheaving), 607 Sim s, 647 Shear, 168 Simultaneous congruences, 143 Sheaves, 267 Simultaneous equations model (SEM), 641 Sheaving, 607 just-identified, 642 Sheet, 78, 306 over-identified, 642 Sheff er, 1 under-identified, 642 Shem es, 269 Sinai-Bowen-Ruelle (SBR), 725 Shift operator, 644 Sine, 175 Shifted endogenous variable, 625 Sine function, 174 Shifting parameter, 666 Single unit, 28 Shilnikov theorem, 738 Single-valued function, 431 Shilnikov-Turaev, 748, 749 Singly periodic function, 436 Shilnikov-Turaev-Gavrilov, 748 Singular cardinal, 15, 578 Shinai, 733 Singular complex, 286 Shinai theorem, 733 Singular homology, 263, 549, 550 Shock, 644, 647 Singular metric projective plane, 158 Short exact sequence, 554, 605 Singular n-cochain, 317 Short memory process, 650 Singular n-simplex, 286 Short term Fourier transformation (STFT), 666 Singular point, 428 Shortest path between two points, 373 Singular point of algebraic curve, 336 Shrinking, 249 cusp, 336 Shrinking of a closed curve to a point, 197 isolated point, 336 Sierpinski carpet, 731 no de, 336 Sierpinski gasket, 731 Singular point, index of a, 412 Sieve, 607, 608 Singular solution, 381 Sieve of Eratosthenes, 147 Singular theory, 287 Sigma algebra, 18, 618 Singular values, 727 Sigma field, 18, 618 Singular values of a matrix, 727 Signal analysis, 665 Singularities of analytic functions, 433 Signal reconstruction, 665 Singularity, 421 Signal synthesis, 665 cusp, 702 Signature, 138, 510, 511, 513 degenerate, 699 Signature (disambiguation), 511 hyperbolic, 699 Signature (in logic), 511 non-degenerate, 699 Signature (of a knot), 512 non-hyperbolic, 699 Signature (of a manifold), 512 of a complex function, 429 Signature (of a metric), 512 of analytic functions, 433 Signature (of a permutation), 511 of holomorphic functions, 428 Signature (of a prime), 511 Singularity of holomorphic functions, 428 Signature (of a structure), 511 Sink, 348, 717 Signature (of a symmetric or Hermitian matrix), 511 Sinusoidal function with harmonics, 726 Signature (of an algebraic system), 511 Site, 569, 607, 608 Signature (of quadratic form), 511 Skeleton, 254, 260, 295, 535 Signature (type), 510 Skew-symmetric, 138 Signature of a permutation, 87, 105 Skew-symmetrization, 138 Signature of a transposition, 87 Skewness test, 643 Similar circles, 166 Slack variable, 669 Similar figures, 165 Slater regularity condition, 673, 674 Similar triangles, 166 Slice category, 585 Similarity, 149, 165, 173, 193 Smale theorem, 738 direct, 452 Smale-Birkhoff theorem, 709 Similarity (similitude), 165 Small étale site, 568 Similarity in geometry, 165 Small category, 302, 568, 582, 585 Similarity type, 511 Small circle, 179 Similitude, 165 Small class, 16, 577, 578, 585 Similitude center, 166 Small colimit, 585 Similitude group, 149 Small initial variation, 717 Simple algebra, 530 Small limit, 585 Simple autocorrelation function, 643 Small perturbations, 702 Simple chain, 260 Small sets, 16, 577 Simple closed curve, 392 Smallest convex set, 250, 252, 255 Simple convergence, 234 Smallest element, 9 Simple curve, 214 Smash product, 312 Simple dipath, 259 Smash product of pointed space, 312 Simple extension, 123 Smith normal form, 296 Simple graph, 259 Smith normal form theorem, 296 Simple isolated singular point, 428 Smooth embedding, 544, 545 Simple isolated singularity, 433 Smooth function, 416, 546 Simple iterative method, 340 Smooth functions on manifolds, 546 Simple linear correlation coefficient, 625 Smooth manifold, 540, 541 Simple order, 45 Smooth map, 546 Simple radical extension, 129 Smooth map (smooth function), 546 Simple regression model, 626 Smooth maps between manifolds, 547 forecasting, 627 Smooth morphism, 566 geometric interpretation, 628 Smooth scheme, 564

809 Index

Smooth structure, 541 Spring-mass system, 377 Smooth surface, 415 Oscillatory motion, 377 Smooth unramified morphism, 569 Spurious regression, 648 Smooth variety, 569 Square matrix, 109 Smoothly universal, 544 Square matrix of order n, 109 Snake lemma, 554 Square wave, 668 Sobolev space, 391 Square-integrable periodic functions, 245 Soh, 175 Squaring the circle, 75 Solow model (Pascal distribution), 641 SS (sum of squares), 627 Solow-Swan model, 753, 758 SSE (sum of squares explained), 627 Solution, 314 Ssingularity, 441 periodic, 717 SSR (sum of squares of residuals), 627 stable, 686 SST (total sum of squares), 627 Solvable group, 89 Stability Solving equation ∂u/∂z¯ = v, 480 asymptotically stable, 719 Sort, 512 Kolmogorov (invariant tori preservation in Hamiltonian), 718 Sotomayor, 711 Lagrange stability, 718 Source, 348, 598, 600, 717 Lyapunov stability, 718 South pole, 271 orbital stability, 719 Space Poisson stability, 718 B aire, 232 Structural stability, 718 compact, 223 Stability Boundaries, 748 complete, 230 Stability in the sense of Lyapunov, 757 contractile, 249 Stability of periodic orbits, 719 contractile arcwise connected, 249 Stability of steady states, 719 Hausdorff, 199 Stability problem of a periodic solution, 717 Hilbert, 199 Stability test by recursive regression, 631 2 L ([a, b], K), 244 Stability tests, 631 l2(K), 244 Stability theory, 718 L2, 392 Stabilizer, 306 Lp, 392 Stable mapping (function), 233 additively, 40 m etric, 50 multiplicatively, 40 complete, 67, 215, 228 Stable (unstable) eigenspace, 697 of 2π-periodic functions, 245 Stable fixed point of Banach, 371 in the sense of Lyapunov, 687 of Hausdorff, 199 Stable in the sense of Lyapunov, 693 of Hilbert, 371 Stable part, 32 pre-Hilbert, 370 Stable part by a map, 30 semi-metric Stable part for a law, 40 separated, 230 Stage (or domain of variation), 581 semimetric, 230 Stage of definition, 581 Sobolev, 391 Stagewise regression, 633 T 2, 199 Stalk, 267, 552, 607 topological, 50 Stalk (of a presheaf), 552 totally disconnected, 226 Stalk (of a sheaf), 552, 607 uniform Standard étale, 569 complete, 231 Standard étale morphism, 569 Space average, 725 Standard deviation, 616, 621 Space Cn, 440 Standard function, 499 Space diagonal, 299 Standard n-simplex, 286 Space L2([a, b], K), 244 standard n-simplex, 271 Space l2(K), 244 Standard normal distribution, 617 Space of 2π-periodic functions, 245 Standard Zariski cover, 567 Space of all distributions, 496 Star, 152, 294 Space of finite type, 318 Star convex, 196 Space with no holes, 545 Star-shaped open set, 498 Span (hull), 250, 255 State variable, 684, 685 Span„ 601 State-space model, 684 Sp ec, 531 Static model, 648 Spec (as a set), 529 Stationarity, 643 Spec (as a sheaf), 529 Stationarity tests Spec (as a topological space), 529 augmented Dickey-Fuller test (ADF), 644 Spec(R), 529 Dickey-Fuller test (DF), 644 Spec(Z), 529, 572 Stationary Gaussian process, 651 Special chain, 506 Stationary series Special linear group, 131, 417 (i.i.d.), 643 (n.i.d.), 643 Special linear group SLn(q), 417 with memory, 643 Special linear group SLn(R), 417 Special orthogonal group SOn, 172, 191 Statistic Special orthogonal group SOn(q), 191 of Jarque-Bera, 643 Special unitary group, 131 Statistical distribution, 620 Spectral energy, 654 Statistical induction, 625 Spectral interpretation of Fourier transform, 662 Statistical inference, 625 Spectral radius, 494 Statistical methods, 622 Spectral radius formula, 494 Statistical models, xviii, 624 Spectral theorems, 24 Statistical test, 629 Spectral topology, 529 Statistics, xviii, 615 Spectral value, 133 Steady state, 714 Spectrum, 24, 269, 466, 494, 662 Step function, 350, 658 of aperiodic signal, 727 lower-bound, 350 of chaotic signal, 727 upper bound, 350 of periodic signal, 726 Stepwise regression, 633 Spectrum of a bounded operator, 24 Stereographic projection, 78, 179, 271, 422, 482, 687 Spectrum of a ring, 270, 529 Stereographic retraction, 312 Spectrum of function with f1/f2 irrational, 726 Stieltjes integral, 369, 375 Spectrum of function with f1/f2 rational, 726 Stieltjes transformation, 657 Spectrum of periodic signal, 726 Stirling formula, 90 Spectrum of polynomial ring, 270 Stirling number, 90 Spectrum of quasiperiodic signal Stochastic, 643 of quasiperiodic signal, 726 Stochastic assumptions, 628 Speculation effect, 760 Stochastic matrix, 616 Speed of convergence, 340 Stochastic optimization, 670 Sphere, 184, 539, 708 Stochastic process, 616, 623, 642 Sphere model, 179 Stochastic programming, xviii, 669, 674 Sphere of curvature, 396 Stochastic variable, 616 Sphere theorem, 420 Stokes and Ostrogradsky formulas, 138 Sphere with handle, 432 Stokes formula, 411, 450 Spherical arc, 396 pro of, 452 Spherical geometry, 179 Stokes formula for the rectangle, 411, 448 Spherical point, 419 Stokes theorem, 316, 363, 411 Spherical triangle, 180 closed form, 364 Spinning diffeomorphism, 709 exact form, 364 Spinor, 360 general form, 363 Spiral, 717 Stokes theorem (in five dimension), 696 Spiral source, 717 Stokes theorem applications to tubular region, 696 Split exact sequence, 303, 554 Stone-Weierstrass theorem, 236, 245 Split monomorphism (coretraction), 598 Straight line, 150 Split polynomial, 113 Straightedge, 129 Splitting field, 124 Straightedge and compass, 129

810 Index

Strange attractor, 724, 730 Support of a distribution, 496 Strange attractor (Ruelle-Takens), 733 Support of a family, 40 Strange attractor in the sense of Ruelle-Takens, 733 Support of a map, 443 Strange attractors, 732 Support of a measure, 442 Stranger, 111 Support of distributions, 668 Stratified sampling, 615 Support of i-chain, 506 Strict subset, 7 Supremum, 45, 46 Strictly concave function, 678 Surface, 255, 398, 399 Strictly convex function, 679 hyperbolic Riemann, 415 strictly less than, 44 local parametric representation, 414 Strong deformation retract, 301, 573 locally orientable, 400 Strong deformation retraction, 301 of Riemann, 415 Strong extrema, 373 Orientability and topological invariant, 256 Strong induction, 51 orientable, 400 Strong law of large numbers (Kolmogorov), 620 sm o oth, 415 Strong recurrence, 51 Surface group, 298 Strong Stable Manifold, 748 Surface integral, 360, 362, 411 Strongly divergent sequence, 322 Surface of absolute values of complex functions, 423 Strongly inaccessible cardinal, 15, 578 Surface of revolution, 188, 360, 402 Structural assumptions, 628 ellipsoid of revolution, 188 Structural equations, 641 hyperboloid of revolution, 188 Structural form of VAR representation, 646 paraboloid of revolution, 188 Structural stability, 702, 703, 718 Quadrics of revolution, 188 Structural stability (robustness), 723 Surface with boundary, 255 Structural stability of linear systems, 703 Surfaces in R3, 414 Structural stability of the locking, 744 Surgery, 373 Structurally stable differential equation, 723 Surjection, 520 Structurally stable vector field, 703 Surjective embedding (isomorphism), 520 Structure, 38, 511, 512 Surjective homomorphism, 520 algebraic, 81 Surjective homomorphism (epimorphism), 520 topological, 50 Surjective map, 29, 31, 519 Structure (algebraic system), 511 Suspension, 311, 692, 693 Structure homomorphism, 520 Suspension of a diffeomorphism, 693 Structure map, 593 Syllogism rule, 2 Structure morphism, 563 Sylvester formulas, 618 Structure of lags, 640 Symb ol, 1 Structure of Q, 67 Symb ol h , 347, 429 Structure of the set of reals, 38 Symbol sequences, 707 Structure of the uniform convergence on F E , 234 Symbolic dynamics for horseshoe diffeo., 707 Structure sheaf, 270 Symbolic logic, xvii Structure sheaf , 552 Symmetric, 40 Structure-preserving bijection, 258 Symmetric bilinear form, 406 Structure-preserving map, 545 Symmetric braided monoidal category, 587 Structures, xvii, 37 Symmetric difference, 13, 33, 93 Structures (logic), 512 Symmetric group, 86 Structures on R, 321 Symmetric monoidal category, 587, 588 Student law, 625, 629 Symmetric sesquilinear form, 242 Student ratio, 629 Symmetrizable, 40, 82 Student t-distribution, 617 Symmetry, 162 Student’s table, 625 Symplectic diffeomorphism, 695 Sub-mean-value property, 466 Symplectic manifold, 695 Subadditivity, 20 Synchronization of frequencies, 741 Subalgebra, 375 Synchronization of oscillators, 743 Subarc, 448 Syntactic method, 2, 3 Subbasis of a topology, 200 σ-system (Σ-system , Ω-system), 511 Subcategory, 257, 265, 272, 584 System of class representatives, 28 Subcomplex, 290, 535 System of cogenerators, 581 System of congruences, 97 Subcritical bifurcations of fixed points, 734 System of differential equations, 383, 684 Subfield, 38, 69, 99 System of equations, 106 Subgraph, 258 System of first-order differential equations, 384 Subgroup, 38, 83, 599 System of generators, 581 Subgroup generated by a part, 83 System of higher-order differential equations, 384 Subgroup generated by a set, 83 System of linear equations, 107 Subgroup of Z, 59 System of linear first-order diff. equations, 384 Subharmonic, 493 System of propositional formulas (axioms), 2 Subharmonic bifurcation, 718 System of relations, 38 Subharmonic distribution, 492 System of structural equations, 641 Subharmonic function, 490, 491 System of types, 14 Subharmonic instability, 734 Subharmonicity, 492 T-algebra, 593 Sublattice, 22 t-distribution, 617 Sublinear function, 20 T-periodic , 714 Sublinear functional, 20 T0,T1,T2,T3,T4-axioms, 206 Submanifold, 543 T0,T1,T2-axioms, 206 Submanifold atlas, 544 T1-space, 247 Submanifold property, 544 T2-axiom, 199, 206 Submodule, 38, 101 T2-space, 199, 206, 247 Subobject, 580 T2-space (Hausdorff space), 207 Subobject classifier, 581, 608 T3-axiom, 207 Subproposition, 1 T3-space (regular space), 207 Subring, 38, 93, 99 T4-axiom, 207 Subroup, 83 Table of indefinite integrals, 353 Subscript notation, 30 Table of integration rules, 356 Subsequence, 221, 322 Table of primitives, 353 Subset, 7, 11 Table of random numbers, 624 Subset relation, 13 Table of variance analysis, 630 Subspace topology, 538 Tables of Fourier transformation, 661 Subspace topology (relative topology, induced topology, trace topology), 538 Tables of random numbers, 623 Substitution principle, 289 Takens, 712, 713, 733 Substructure, 38, 99 Takens theorem, 416 Substructures, 99 Tangent, 175, 393 Substructures (subring, subfield), 99 Tangent angle, 163 Subsystem of relations, 38 Tangent developable, 396 Successive derivatives, 330 Tangent developable of a curve, 396 Successive integrations, 356 Tangent hyperplane, 343 Successive trials, 619 Tangent linear map, 416 Successor, 15, 47, 48, 51, 322, 578 Tangent of a circle, 163 Sufficient condition, 5 Tangent plane, 399 Sufficient condition (SC), 5 Tangent problem, 328, 329 Sum , 38 Tangent space, 414—416, 689, 708 Sum and product of complex numbers, 70 Tangent vector, 415, 416 Sum of subcomplexes, 290 Tangent vector to a surface, 399 Sum topological space, 202 Tangential component, 401 Sum topology, 202 Tangle of curves, 721 Summable family, 240 Target, 600 Supercritical Andronov-Hopf bifurcation, 749 Tarski’s axiom, 16 Supergraph, 258 Tarski-Grothendieck set theory (TG), 15 Superposability, 29, 161 Tautological propositional formula, 2 Supplementary angles, 162 Tautology, 1 Support, 40, 108, 287 Taylor expansion, 701

811 Index

Taylor formula, 114 Cauchy integral, 425, 429 Taylor polynomial, 332 Cauchy integration criterion, 355 Taylor polynomial theorem, 332 Cauchy theorem (homology version), 475 Taylor remainder, 332 Cauchy theorem (homotopy version), 470 Cauchy form, 332 Cauchy-Goursat, 453, 461 Lagrange form, 332 Cauchy-Kowalewska, 389 Taylor series, 333, 426 Cauchy-Pompeiu formula, 456 Tchebychev (Chebyshev, Tchebicheff, Ceby˘sev,˘ Tschebyscheff, Chebishev), Cayley-Hamilton, 134 75, 339 center manifold, 700 Tear, 194 central limit, 617 Tensor, 105, 138, 405—407, 415 C hinese, 97 relative, 408 Chinese remainder, 97 skew-symmetric r-times contravariant, 138 chordal, 167 Tensor algebra, 406, 407 circle quadrature, 131 Tensor analysis, 407, 408 closed graph theorem, 241, 372 Tensor bundle, 408 commutative group for composition law, 156 n Tensor calculus, 407 com pact of R , 197 Tensor category, 586 compact set, 209 Tensor contraction, 407, 410 compact support, 288 Tensor density, 408 compactness criterion of metrizable space, 227 even, 408 completion of a metric space, 229 o dd, 408 Complex Hahn-Banach, 25 Tensor differentiation, 407 composition rule, 330 Tensor field, 406—408 congruence, 163 Tensor of type (0, 1), 406 connected image, 203 Tensor of type (0, 2), 406 connected space, 203 Tensor of type (1, 1), 406 constructibility by straightedge-compass, 130 Tensor of type (p, q), 406 convergence criterion, 340 Tensor on a vector space, 408 covering homotopy theorem, 306 Tensor powers, 525 curl, 348, 412 Tensor powers and braiding, 525 cyclic decomposition, 136 Tensor product, 407, 586 degree, 122 Tensor quantity, 407 D enjoy, 692 Tensor set, 407 derivability, 442 Tensor space, 407 Desargues (affine version), 155 Tensor space of type (α, β), 407 Desargues (little), 157 Tent map, 725 Desargues (projective version), 155 Term of degree k, 108 Diagonalization conditions, 134 Term of degree n, 290 dimension axiom, 287, 312, 317 Terminal morphism, 580 D ini, 234 Terminal object, 310, 580 divergence, 364, 412 Terminal object , 580 divergence (Gauss), 348 Terminal property, 580 Douady-Oesterlé, 731 Ternary relation, 28, 160, 161 Douady-Oesterlé (for diff. equations), 731 Test dual universal coefficients, 318 C how , 635 Eilenberg-Steenrod, 302 Kurtosis (flattening), 643 Eilenberg-Steenrod (extension), 302 normality, 643 Eilenberg-Zilber, 305 Skewness (asymmetry), 643 entire series expansion and Taylor series, 333 Test function, 668 Euclid, 56, 169 Tetragon, 154 Euclid lemma, 57, 60 Tetrahedron, 251, 255 Euclid lemma in Z, 60 T hales, 163 Euclidean division, 55, 59 T heorem Euclidean division in Z, 59 group of affine collineations: G, 172 Euler, 96, 167 A b el, 129 Euler criterion, 144 about angles, 162 Euler displacement, 186 acyclic free chain complex, 303 Euler-Lagrange, 83, 84 addition, product, quotient rules, 330 exact triangle, 291 Afraimovich-Shilnikov, 739 excision, 296, 317 Alembert, 224 extension, 228 Alembert-Gauss, 77 extreme value, 209 Alexander-Whitney, 319 factorization, 57 Alexandroff, 225 Fatou, 488 Andronov-Hopf, 735 Fermat, 89, 93, 96 Andronov-Pontryagin, 724 Fermat (little), 89 Andronov-Witt, 719 Fermat’s last, 93, 96 anti-matching, 153 Fermat’s little, 96 anticommutativity, 319 field by two ideals, 101 any Euclidean ring is principal, 142 field of complex numbers is algebraically closed, 115 any finite extension is algebraic, 124 field with characteristic zero, 125 Ascoli, 236, 442 finite Abelian group, 89 Ascoli (for a metric space), 236 finite simple extension, 123 Ascoli (general case), 236 finite simple extension (Steinitz), 123 aureum , 144 five lemma, 291 B ézout, 60 fixed-point, 229 Bézout (for polynomials), 111 fixed-point with parameter, 229 Baire category, 232, 724 Floquet, 699, 716 Banach, 241, 391 flow-box, 691 Banach fixed point, 372 Fredholm, 374, 390 Banach-Alaoglu, 442 Freudenthal suspension, 315 Banach-Steinhaus, 241, 371 Frobenius, 417 Banach-Tarski paradox, 17 Fundamental theorem of Algebra, 77 basis of a vector space, 103 fundamental theorem of arithmetic, 57 B ayes, 619 fundamental theorem of arithmetic (for polynomials), 111 Bessel inequality, 244, 653 fundamental theorem of integral calculus, 352 binomial, 90, 92 Gauss, 57, 60, 348 biquadratic reciprocity, 145 G auss (in Z), 60 Birkhoff, 718 genus, 145 Birkhoff (ergodic), 725 G iraud, 609 Birkhoff HSP, 514 golden (Gauss), 144 Birkhofff, 514 gradient, 343, 412 bisectors, 153 Granger representation, 649 Blakers-Massey, 315 Green, 316, 348, 364 Block-Guckenheimer-Misiuriewicz, 733 Green-Riemann, 363 Bolzano-Weierstrass, 322, 323 Grobman-Hartman, 723 Boolean prime ideal (BPI), 19, 23, 24 Grobman-Hartman (for discrete syst.), 723 B orsuk, 473 Grothendieck, 567, 570 Borsuk-Ulam, 294, 320 groupoid, 280 Brouwer, 214, 472 Gysin sequence, 320 Brouwer and Kakutani fixed-point, 674 Hadamard three circles, 491 Brouwer fixed point, 215, 677 Hadamard-Cartan & Gromoll-Meyer, 420 C is algebraically closed, 115 Hadamard-Perron, 720 Cantor, 31, 228 Hadamard-Perron (for discrete system), 723 Cantor-Schröder-Bernstein theorem, 36 Hahn-Banach, 24, 45, 236, 238 Carathéodory, 485 Hahn-Banach (geometric form), 26 Casorati-Weierstrass, 464 Hahn-Banach separation, 26 C auchy, 456 hairy ball, 294, 412 Cauchy formula (homology form), 475 ham-sandwich, 320 Cauchy formula (homotopy form), 471 Harnack inequalities, 486

812 Index

Hartman-Grobman, 697 Palis-Smale, 724 Hartman-Grobman (for diffeomorphism), 697 Pappus-Pascal (affine version), 155 Hartman-Grobman (for flow), 697 Pappus-Pascal (projective version), 155 Hartogs, 494 Parseval formula, 244 heights, 153 particular irreducible polynomial (Abel), 121 Heine, 223, 224 Peano existence, 386 homoclinic saddle connection, 747 Peixoto (I) (for vector field, flow), 704 homomorphism, 88 Peixoto (II) (for vector field, flow), 704 homothetic circles, 166 Peixoto (III) (for diffeomorphism), 705 homothetic triangles, 166 period three implies chaos, 733 homotopy axiom, 288, 317 perpendicular bisector, 153 homotopy axiom for pairs, 292 perpendiculars (Hjelmslev), 152, 153 homotopy extension, 301 Picard-Lindelöf, 714 homotopy sequence of a fibration, 314 Poincaré, 412 Homotopy sequence of a pair, 314 Poincaré Bendixson, 720 homotopy sequence of a weak fibration, 315 Poincaré recurrence, 718 H opf, 320 Poincaré recurrence (Poincaré return), 718 Hopf bifurcation, 712 Poincaré return, 718 Hopf recurrence, 718 Poincaré-Bendixson, 690 H SP, 514 Poincaré-Brouwer, 412 Hurewicz, 289, 315 polynomial division, 120 Hurewicz (1), 289 p ower, 167 Hurewicz (2), 315 primary decomposition, 136 Hurewicz (absolute version), 315 projection, 21, 26, 242 Hurewicz (relative version), 315 projection on a complete convex set, 242 Hurwitz, 477 projection on a complete vector subspace, 243 immersion, 213 Projective Nullstellensatz, 533 insets and invariant manifold, 709 propositional calculus , 1, 2 integral domain by prime ideal, 100 Puppe sequence, 313 integral mean-value, 351 Pythagoras, 242 intermediate value, 203, 226, 326 Pythagorean, 169, 175, 178 invariance of dimension, 294 quadratic reciprocity, 144 invariance of domain, 294 quadratic reciprocity law, 144 invariance of integral by homotopy, 470 quadratic residue, 144 invariant manifold, 697 radical or power center, 167 invariant manifold (for diffeomorphism), 697 recurrence, 51 invariant manifold (for flow), 697 reduction, 297 inverse function, 390 regular T1-space, 207 inverse mapping, 568 representation, 243 irreducible polynomial, 121 residue, 430, 476 irreducible prime element, 142 restriction rule, 329 isomorphism, 89 R icci, 409 Jensen formula, 463, 492 Riemann conformal representation, 439 Johnson, 494 Riemann criterion, 350 Jordan, 473, 474 Riemann mapping, 483 Jordan canonical form, 136 Riemannian geometry (fundamental th.), 418 Jordan curve, 214, 294, 504 R iesz, 375 Jordan-Brouwer, 214 Riesz decomposition, 492 Jordan-Brouwer separation, 294 Riesz representation, 21, 375, 442 Künneth formula for cohomology, 318 Rolle, 331, 332 Künneth theorem, 305 roots and zeros of a polynomial, 112 Kakutani (fixed point), 677 roots of a polynomial, 112 Kelvin-Stokes, 364 roots of a polynomial of degree n, 121 Kolmogorov-Arnold-Moser (KAM), 724 Rouché, 477, 504 Kondrakov, 391 Runge, 478, 479 Kuhn-Tucker, 673, 682 Schoenflies, 215 Kuratowski, 263 Schwartz, 467 Kuratowski-Pontryagin, 263 Seifert-van Kampen, 297, 298 L’Huilier, 180 separability criterion for a polynomial, 125 Löwenheim-Skolem, 2 separating hyperplane, 26 Lagrange, 83, 86 Serre, 314 Lagrange interpolation polynomial, 112 Sharkovskii, 733 Lagrangian (generalized), 681 Sharkovskii -1964, 733 least upper bound, 66 Sharkovsky, 733 Lebesgue differentiation, 488 Shilnikov, 738 Ledrappier, 731 Shinai, 733 Lefschetz, 304 signature of a permutation, 87 Lefschetz fixed point, 304 signature of a transposition, 87 Leray, 296 similar triangles, 166 lifting criterion, 307 simple and multiple roots of a polynomial, 112 lifting lemma, 306 simplicial approximation, 295 lim it, 620 Sm ale, 738 linear extension, 103 Smale-Birkhoff, 709 linear map and matrix associated, 104 Smirnow-Nagata-Bing (fundamental th.), 211 Liouville, 77, 463, 465, 695, 716 Smith normal form, 296 Liouville (vector version), 465 sphere, 420 Liouville boundedness, 433 Stokes, 316, 363, 411 Littlewood, 493 Stokes (closed form), 364 long exact sequence, 317 Stokes (exact form), 364 Lusternik-Schnirelmann, 294 Stokes (general form), 363 Mac Lane coherence, 586, 587 Stone-Weierstrass, 236 Malgrange, 711 Stone-Weierstrass , 245 Malgrange-Ehrenpreis , 669 Stone-Weierstrass (for complex functions), 236 M ather, 705 Takens, 416 Mayer-Vietoris, 292 Taylor polynomial, 332 Mayer-Vietoris (for reduced homology), 292 T hales, 163 mean value, 331 theory of patches (fundamental th.), 405 mean value inequality (for polynomials), 115 three circles, 491 medians, 153 T ietze, 297 M enger, 262 Tietze-Urysohn, 442 M enger n-arc theorem, 262 Tikhonov, 209 Mergelyan, 479 total probability, 619 metrization, 210 transport of intervals, 326 Meusnier, 402 Turan, 262, 263 Mittag-Leffer, 481 Tychonoff, 22, 45, 209, 224, 579 modified Jordan form, 758 Tychonov, 224 monic irreducible polynomial, 121 undecidability, 3 monodromy, 427 unique factorization, 57 M ontel, 458 unique polynomial associated with an algebraic element, 122 M orera, 461 unique-factorization, 142 Nagata-Smirnov (metrization), 212 uniqueness, 386 Neimark-Sacker, 737 uniqueness of the norm, 494 no-retraction, 472 unisolvence (for polynomial interpolations), 113 non constructibility (straightedge comp.), 130 universal coefficients for cohomology, 318 normal form (differential equation), 698 universal coefficients for homology, 304 numeration in base b, 56 Urysohn, 207, 211 of well-ordering, 17 Urysohn (metrization), 212 open mapping, 241, 372, 472, 477, 482, 537 van Kampen, 297, 298 Ostrogradsky-Stokes, 363 variation of constants formula, 716 Ostrowski, 145 vector space norm, 371

813 Index

Vesentini, 494 Topologically equivalent flows, 692 Wedderburn, 114 Topologically equivalent sets, 194 Weierstrass, 481 Topologist’s sine curve, 204 Weierstrass convergence, 458 Topology, xvii, 20, 27, 193, 218 Weierstrass product, 433 associated with semi-metric structure, 230 Whitehead, 315 com parison Whitney, 415, 416 coarsest, finest, coarser, finer, 199 Whitney graph, 259 induced, 199 Wiener-Khinchin, 726 m etric, 199 W ilson, 98 p-adic, 79 Yoneda lemma, 302 relative, 197 Young (1), 731 Topology change, 733 Young (2), 731 Topology engendered by maps, 201 Zermelo, 36, 47 Topology generated, 199 zeros of a polynomial, 112 Topology generated by a family of parts, 218 Zorn, 45, 47 Topology of bands, 199, 204 Theorem of homothetic triangles, 166 Topology of the complex plane, 472, 497 Theorem of similar triangles, 166 Topology on a set, 538 Theorems about angles, 162 Topos, 569, 608 T heory Tor, 304, 560 of approximation, 338 Tor group, 304 of cointegration, 626 Torsion, 304, 394 of differential equations, xvii Torsion coefficient, 257, 296 of interpolation, 339 Torsion formula, 395 Theory of analytic functions, 421 Torsion group, 257 Theory of curves, 214 Torsion product, 304 Theory of differential equations, xvii Torsion sheaf, 571 Theory of patches, 404 Torsion tensor, 418 fundamental theorem, 404 Torus, 275, 402 Theory of simplicial homology, 257 Torus T 2, 730 Theory of singular homology, 257 Total curvature, 403 Theory of types, 13, 16 Total degree, 117 Three circles theorem, 491 Total order, 9, 21, 36, 44, 46 T i-space, 207 Total order relation, 34 Tietze theorem, 297 Total order structure of N, 63 Tietze-Urysohn theorem, 442 Total orthonormal family, 244 Tikhonov space, 207 Total orthonormal system, 244, 245 Tikhonov theorem, 209 Total probability, 619 Tile, 209, 355 Total probability theorem, 619 T iling, 447 Total space, 306, 560, 562, 574 Time average, 725 Total variability (SST), 627 Time continuous system, 714 Totality, 21, 44, 579 Time discrete system, 714 Totally disconnected topological space, 226 Time series, 642 Totally E-acyclic, 303 Time series analysis (TSA), 642 Totally ordered set, 9, 21, 44—46 Time-dependent normal form, 699 Totally ordered set (Total order), 44 Time-lag model, 625, 639 Totally well ordered field Time-series, 624 Archimedean , 78 Time-series cross-sectional data (TSCS), 633 Totient function, 57, 68, 96 Tivialization of line bundle on a torus, 269, 562 Trace, 201, 304, 717 Toa, 175 Trace map, 432 Tobin model, 753 Trace topology, 538 Tobin monetary growth model, 760 Traces, 194 Tolerance threshold, 622 Tractrix, 177 Topol. equivalence replaces top. conjugacy, 696 Tractroid, 177 Topological étale site, 610 Transaction effect, 760 Topological and continuous invariants, 195 Transcendantal extension, 123 Topological basis, 199, 308, 543 Transcendental elements of a field, 122 properties, 199 Transcendental function, 433 Topological completion, 78 Transcendental number, 78, 122 Topological conjugacy, 692, 696 Transcendental numbers, 117 Topological conjugacy of discrete systems, 723 Transcritical bifurcation (normal form), 733 Topological direct sum, 243 Transcritical bifurcations, 735 Topological dual, 238 Transfinite induction, 47, 49 Topological embedding, 544, 575 Transformation Topological entropy, 727 elliptic, 439 Topological equivalence, 416, 696, 722, 733 hyperbolic, 439 Topological equivalence of diff. equations, 722 loxodromic, 439 Topological graph, 258 parabolic, 439 Topological group, 38, 66, 283 Transformation group, 149 Topological homeomorphism, 200 Transformation matrix, 758 Topological invariance, 248 Transition function, 414, 542 Topological invariance of dimension, 540 Transition map, 414, 542 Topological invariant, 193, 195 Transition map (transition function), 542 Topological isomorphism, 200 Transition matrix, 623 Topological K-theory, 269, 562 Transition probabilities, 623 Topological linear space, 20 Transitions to Chaos, 738 Topological manifold, 539, 540 cascade of period-doublings, 738 Topological manifold are locally compact, 540 destruction of a torus, 738, 739 Topological manifolds are paracompact, 540 Global homoclinic bifurcation, 738 Topological map, 193, 194, 200 intermittency, 738 Topological n-dimensional manifold, 541 Transitive G-set, 306 Topological n-manifold, 539 Transitive G-space, 306 Topological quotient space, 38 Transitivity, 7 Topological space, 27, 50, 197, 198, 217, 537 Transitivity rule, 2 axiomatic definition, 198 Translation, 154, 161, 162, 164 comparison, 200 Translation functor, 591 defined in terms of neighborhoods, 197 Transmission medium, 661 defined in terms of open sets, 198 Transpose, 106 example, 198 Transpose conjugate matrix, 136 in terms of neighborhoods, 537 Transposition, 86 in terms of open sets, 537 Transvection, 168 T0,T1,T2,T3,T4-axioms (separation), 207 Transversal, 162 totally disconnected, 226 Transverse homoclinic point, 709, 723, 738 triangulable, 253 Trapezium, 355 Topological space in terms of neighborhoods, 537 Trapezium (approximation) method, 355 Topological space in terms of open sets, 537 Trapezoidal formula, 656 Topological structure, 38, 50, 65, 218 Tree, 260, 297, 309 transfer, 201 Trees, 619 Topological structure of Q, 65 Trefoil curvE, 336 Topological structure of R, 321 Trend, 650 Topological subspace, 38, 200, 201 Trial, 616 Topological theorem of complex numbers, 78 Triangle, 154, 175 Topological transitivity (TT), 732 Triangle axiom, 20 Topological vector space, 20, 38 Triangle centroid, 167 Topologically conjugate, 697, 723 Triangle of signed Stirling numbers of the first kind, 90 Topologically conjugate diffeomorphisms, 690 Triangulable topological space, 253 Topologically conjugate hyperbolic linear diffeomorphism s, 697 Triangular matrix, 135 Topologically equivalent, 194, 197, 697, 722 Triangular simplex, 255 Topologically equivalent distances, 217 Triangular system, 642

814 Index

Triangulated category, 591 Uniform topology, 230 Triangulated polygon, 298 Uniformity, 230 Triangulation, 250, 251, 253, 255, 294, 295 Uniformly approachable map, 338 Trigonalization, 135 Uniformly continuous function, 327 Trigonometric circle, 72 Uniformly continuous map, 227 Trigonometric function, 175, 178, 455 Uniformly convergent sequence of functions, 327 cosecant, 178 Uniformly equivalent distances, 217, 228 cosine, 178 Union, 8, 9, 13 cotangent, 178 Union of open balls, 50, 199 hyperbolic cosine, 178 Union of open sets, 195 hyperbolic sine, 178 Union of regular patches, 399 hyperbolic tangent, 178 Unipolar rectangular impulse, 662 secant, 178 Unique factorization theorem, 57 sine, 178 Unique-factorization domain, 124 tangent, 178 Unique-factorization domain (UFD), 142 Trigonometric identities, 178 Unique-factorization ring, 124, 142 Trigonometric series, 653 Unique-factorization theorem, 142 Trigonometrical function, 174 Uniqueness of distributive lattice, 9 Trigonometry, 174 Uniqueness of the limit, 206 based on a circle, 178 Uniqueness theorem, 386 based on a hyperbola, 178 Unisolvence theorem, 113 Trimetric projection, 174 Unisolvence theorem (for polynomial interpolations), 113 Triple, 593 Unit, 43, 141, 516, 593 Trivial bundle, 269, 561 Unit circle, 159, 214, 717, 748 Trivial divisor, 95, 110 Unit circle of C, 72 Trivial group, 82, 89 Unit element, 42, 43, 515, 516 A blian, 89 Unit jump, 658 cyclic, 89 Unit law, 591 Trivial homomorphism, 282 Unit n-sphere, 539 Trivial morphism, 85 Unit object, 586 Trivial ring, 523 Unit polynomial, 36, 109 Trivial subgroup, 83 Unit ring, 43, 516, 522 Trivial topology, 198 Unit ring (ring with identity, ring with unity), 522 Trivialization, 269, 561 Unit ring (ring with identity, ring with unity, ring with a one, unitary ring, True coefficient, 625 unital ring), 522 True proposition, 1 Unit root, 644 Truncated Taylor series, 685 Unit root test, 644 Truncation, 312, 701 Unit square, 65 Truth value, 1, 2 Unit tangent vector, 393, 394 TS process, 644 Unit vector, 181, 399 TS process (Trend Stationary), 644 Unital Banach algebra, 24 Tube lemma, 299 Unital ring, 43, 516 Turan graph, 262 Unitary divisor, 147 Turan theorem, 262, 263 Unitary group, 131 Turbulence burst, 718 Unitary polynomial, 36, 109 Turbulent phase, 738 Unitary ring, 43, 516, 522 Turing, 3 Unitary tile, 250 Twin primes, 148 U nitor, 586 T w ist, 165 Unity, 42, 515 Twisted K-theory, 562 Universal algebra, 264, 509, 510 Twisted sphere, 155 Universal coefficients, 304 Two point compactification of R, 422 Universal coefficients theorem for cohomology, 318 Two-cycle, 86 Universal coefficients theorem for homology, 304 Two-sided convolution, 662 Universal cover, 574 Two-sided ideal, 22 Universal covering space, 307, 574 Two-sided ideal (bilateral ideal), 100 Universal element, 9 Two-sided test, 617 Universal morphism, 581 Two-sided transformation, 656 Universal properties of diagrams, 589 Two-stage least squares (2SLS), 642 Universal property, 63, 216, 231, 302, 580, 581 Two-tailed test, 617 of completions, 216 Two-value principle, 1 Universal property of the embedding, 91 Two-valued logic, 1 Universal quantifier, 10 Two-valued principle, 4 Universal set, 8, 13, 511 Tychonoff space, 207 Universally catenary ring, 525 Tychonoff theorem, 22, 45, 209, 224, 579 Universally closed morphism, 565 Tychonov theorem, 224 Universally true, 2 Type, 13, 510—512 U niverse U, 577 Type (sort), 512 Univocal relation, 29, 519 Type I error, 617 Unknown error, 626 Type II error, 617 Unramified morphism, 565, 569 Type of algebraic system, 510 Unsatisfiable formula, 2 Type theory, 13, 16 Unstable in the sense of Lyapunov, 694 Types of affine isometry, 187 Unstable separatrix, 747 Types of equilibrium point, 721 Unstable statistical model, 631 Typical property, 724 Upper (Riemann) integral, 350 Upper bound, 45, 46, 53, 490 U-set, 579 Upper box-counting dimension, 730 U-sm all, 579 Upper capacity, 730 U-small set, 579 Upper envelope, 230 Ultra-metric distance, 220 Upper integral, 350 Ultrafilter, 19, 22, 209, 222, 223, 543 Upper limit of sequence, 323 Ultrafilter lemma, 19, 24 Upper pointwise dimension, 731 Ultrametric, 146 Upper semi-continuous function, 233 Umbilic, 402 Upper semi-continuous function (u.s.c.), 490 Umbilic point, 402 Upper set, 23 Umbilical point, 402, 419 Upper triangular matrix, 109, 135 Unary operation, 16, 265, 510 Urelement, 2, 16, 578 Unbiased estimator, 617, 622, 626 Urysohn, 207, 211, 213 Unbiased forcasting, 627 Urysohn metrization theorem, 212 Uncorrelated errors, 626 Usual topology of Cn, 224 Uncorrelated variables, 625 Usual topology of Rn, 224 Uncountable negligible set, 367 Utility maximization under budget constraint, 681 Uncountable product, 217 UVAR (unrestricted vector AR), 647 Uncountable universe, 578 Uncountably infinite set, 18 -valued presheaf, 606 Undecidability theorem, 3 ValeurV d’adherence, 220, 228 Undecidable hypothesis, 35 Valuation, 79, 142, 145 Underlying set, 510, 512 discrete or dense, 145 Underlying set (carrier set), 510 Non-Archimedean, 146 Underlying set (or domain), 512 Valuation of a polynomial, 110 Underlying space, 294 Valuation of n at p, 57 Unfolding, 711 Valued field, 79, 146 Uniform continuity, 227, 327, 424 Van der Pol equation, 735 Uniform convergence, 234, 327 Van der Pol oscillator, 747 Uniform convergence of a sequence, 234 Van der Pol system, 747 Uniform medium, 661 Van Kampen theorem, 297, 298 Uniform norm, 21, 338 Vandermonde matrix, 113 Uniform norm (Chebyshev norm), 338 Vanishing line, 170 Uniform space, 230 Vanishing point, 170 Uniform space (Weil), 230 VAR (vector autoregressive), 646 Uniform structure, 230 VAR model, 646

815 Index

VAR process, 646 Weak ∗ convergence, 488 VA R (p), 646 Weakly embedded submanifold, 544 Variability of residuals (SSR), 627 Weakly inaccessible cardinal, 15, 578 Variable, 1 Wedderburn theorem, 114 dependent, 685 Wedge, 296 dum my, 2 Wedge of pointed spaces, 296 free (in logic), 2 Wedge product, 138, 139, 497 functorial, 3 Wedge product (exterior product), 139 input, 685 Weierstrass approximation, 338 linked (in logic), 2 Weierstrass convergence theorem, 458 output, 685 Weierstrass elliptic function, 435 propositional, 1 Weierstrass elliptic functions, 435 sentencial, 1 Weierstrass eta-function, 435 state, 685 Weierstrass formula, 338 Variable element, 581 Weierstrass functions, 435 Variable to be explained, 626 Weierstrass p-function, 435 Variance, 617, 621 Weierstrass ℘-functions, 435 Variance analysis, 630 Weierstrass product theorem, 433 Variance decomposition, 647 Weierstrass sigma-function, 434, 435 Variance of the error, 626 Weierstrass theorem, 481 Variance-covariance matrix, 633 Weierstrass zeta-function, 435 Variate, 616 Weight of the topological space, 537, 543 Variation of constants formula, 716 Weingarten formulas, 404 Variation of the argument along a path, 471 Weingarten map, 419 Variational derivative, 372 Weingarten surface, 419, 420 Variational equation, 719 Well order, 36, 46 Variational method, 390 Well ordered set, 36, 46 Variety, 264, 513, 529 Well ordering, 36, 46 Variety (overview), 513 Well-founded order, 22, 46, 579 Variety in a category, 264, 593 Well-ordered set, 22, 579 Variety of algebras, 513 Well-ordering principle, 46 Variety of universal algebras, 264, 513, 593 Well-ordering principle (equivalent to axiom of choice), 22, 579 VA R M A , 646 Well-ordering theorem, 17 VARMA (vector autoreg. moving average), 646 Well-powered category, 585 Vect-enriched (symmetric) monoidal category, 586 Weyl lemma, 492 Vector, 150, 161, 181 White noise, 633, 644, 651 Vector addition, 181 White noise process, 643 Vector angles, 75 White’s test, 636 Vector basis, 182 White-noise tests, 643 Vector bundle, 268, 269, 561 Whitehead, 14 Vector density, 408 Whitehead theorem, 315 Vector equation of polar plane, 184 Whitney graph theorem, 259 Vector equation with 2 points, 183 Whitney theorem, 415 Vector equation with 3 points, 184 Whole number, xvii, 51, 62 Vector equation with point and direction, 183, 184 Wiener process, 651 Vector error correction model (VECM), 649 Wiener-Khinchin theorem, 726 Vector field, 294, 410, 411 Wilson theorem, 98 Vector function, 341 Winding number, 430 Vector isometry, 187 Window function, 666 Vector Liouville theorem, 465 Windowed Fourier transformation (WFT), 666 Vector of bits, 32 Wintner-Conti condition, 716 Vector operator, 343 With no divisors of zero, 43, 516 Vector product, 140, 182, 183 With no zero divisors, 43, 516 anticommutativity law, 183 Wolf, goat, cabbage problem, 614 distributivity law, 183 Word, 254 mixed associativity law, 183 word, 519 properties, 183 Work done, 360 Vector resolute, 182 Work-energy principle, 360 Vector space, 20, 38, 43, 44, 81, 101, 103, 370, 517 Wronskian, 382, 383 dimension, 103 norm ed, 370 Xet´ , 569 number of element, 103 Y-scheme, 563 Vector-valued function, 341 Yoneda embedding, 612 Vectorial rotation, 175 Yoneda Lemma, 267, 600 Vectorial structure, 190 Yoneda lemma, 302, 612 Vehulst curve, 638 Young theorem (1), 731 Venn (or Euler) representation, 7 Young theorem (2), 731 Venn diagram, 7 Versal unfolding, 711 Z, 16 Vertex angle, 162 z-plane, 71 Vertex connectivity, 262 z-transformable sequence, 664 Vertex cut, 262 Z-transformation, 664 Vertex method, 669 Zariski cover, 567 Vertex set, 294 Zariski topology, 270, 529, 530, 533, 567 n Vertical composition, 264, 585 Zariski topology on Pk , 533 Vesentini theorem, 494 Zermelo and Zorn theorems, 37 Vinculum, 67, 68 Zermelo axiom of choice, 14 Violation of assumptions, 633 Zermelo set theory, 15, 16 Visual examination of residuals Zermelo theorem, 17, 34, 36 visual examination of residuals, 633 Zermelo-Fraenkel axioms, 16, 22, 579 VMA (vector moving average), 646 Zermelo-Fraenkel set theory, 15, 16, 578 VMA representation, 647 Zero, 118 Volume calculation, 357 Zero chain map, 290 Volume formula, 357 Zero complex, 290 von Neumann hierarchy of sets, 14, 578 Zero element, 42, 43, 515, 516 von Neumann integer, 37 Zero inner product, 21 von Neumann integer zero, 37 Zero map, 556 von Neumann method, 37 Zero morphism, 599 von Neumann ordinals, 37 Zero object, 310, 580 von Neumann regular ring, 526 Zero object (null object), 580 von Neumann regular ring (absolutely flat ring), 526 Zero vector, 521 von Neumann universe, 14, 578 Zero-locus, 532 von Neumann-Bernays-Gödel set theory (NBG), 15 Zeros and local extrema, 334 Zeros of a polynomial, 112 Walras, 677 Zeros of holomorphic functions, 462 Walras law, 677, 759 Zeroth Betti number, 502 Walras model, 753 Zeroth de Rham cohomology vector space, 502 Walrasian equilibrium, 759 ZF, 15, 16 Walsh functions, 667 ZFC, 15, 16, 578 Walsh systems, 667 ZFC axioms, 17 Wavelet family, 666 Zorn lemma, 14, 22, 25, 36, 238, 579 Wavelet moment, 666 Zorn theorem, 45 Wavelet order, 666 Wavelet transformation, 665, 666 Wavelets, 665 Weak embedding, 544 Weak extrema, 373 Weak fibration, 314 Weak fibration (Serre fiber space), 314 Weak law of large numbers (Bernouilli), 620 Weak topology, 300, 535

816